boco.tex
% main:: diploma_main.tex
% by clara Strohm
\section{Bounded cohomology of spaces}

Bounded cohomology is the functional analytic twin of singular
cohomology. It is constructed via the topological dual of the singular
chain complex instead of the algebraic one.  The corresponding norm
for singular cochains is therefore the supremum norm:

\begin{defi}
  Let $X$ be a topological space and $k \in \N.$ For a singular
  cochain $f \in \cosingc{k}{X}{\R}$ the (possibly infinite)  
  \defin{supremum norm} is defined via
  \dindex{supremum norm}%
  \nindex{$r}{$\supn{\,\cdot\,}$}{supremum norm}
  \[ \supn{f} := \sup_{\sigma \in \ssing{k}{X}} |f(\sigma)|.\]
  This induces a seminorm on~$\cohsingc{k}{X}{\R}$ by
  \dindex{seminorm}%
  \nindex{$q}{$\supn{\,\cdot\,}$}{seminorm}
  \[ \fa{\varphi \in \cohsingc{k}{X}{\R}} 
     \supn{\varphi} := 
     \inf \bigl\{ \supn{f} \;\big|\; 
             f \in \cosingc{k}{X}{\R}, \cobou(f) =0, [f] = \varphi
          \bigr\}. 
   \]
  We write
  \nindex{Cob}{$\bchn * X$}{set of bounded cochains in~$X$}
  \[ \bchn{k}{X} 
     := \bigl\{ f \in \cosingc{k}{X}{\R} \;\big|\; \supn{f} < \infty 
        \bigr\}. 
  \]
  for the vector space of \defin{bounded }$k$\defin{-cochains}.
  \dindex{bounded cochain}
  \dindex{cochain!bounded}
\end{defi}

It is easy to see that the coboundary operator on the singular cochain
complex~$\cosingc{k}{X}{\R}$ restricts to bounded cochains, i.e., 
\[\cobou\bigl( \bchn{k}{X}\bigr) \subset \bchn{k+1}{X}.\]
Thus $\bchn{*}{X}$ is a cochain complex.

\begin{defi}
  Let $X$ be a topological space and $k \in \N.$ The $k$\defin{-th
  bounded cohomology group of}~$X$ is defined by
  \dindex{bounded cohomology}
  \sindex{cohomology!bounded}{bounded cohomology}
  \nindex{Hob}{$\bch * X$}{bounded cohomology of~$X$}
  \[ \bch{k}{X} := 
                       \frac{\ker\bigl(
                                \cobou|_{\bchn{k}{X}} \colon
                                \bchn{k}{X} \longrightarrow \bchn{k+1}{X}
                             \bigr)}
                            {\im\bigl(
                                \cobou|_{\bchn{k-1}{X}} \colon 
                                \bchn{k-1}{X} \longrightarrow \bchn{k}{X}
                              \bigr)}.
     \qedhere\]
\end{defi}

As for the simplicial volume, it is not immediately clear from the
definition that there are spaces whose bounded cohomology does not
vanish. An example of such a space is $S^1 \lor S^1$
(Example~\ref{bcexcex}).

\begin{defi}
  Let $X$ be a topological space and $k \in \N.$ The supremum norm 
  on~$\bchn{k}{X}$ induces a seminorm on~$\bch{k}{X}$
  by
  \dindex{seminorm}%
  \nindex{$q}{$\supn{\,\cdot\,}$}{seminorm}
  \[ \fa{\varphi \in \bch{k}{X}} 
     \supn{\varphi} := \inf \bigl\{  \supn{f} \;
                            \big|\; f  \in \bchn{k}{X}, \cobou(f) = 0,
                                    [f] = \varphi
                            \bigr\}. \qedhere\] 
\end{defi}

Except for some special cases it is not known whether this seminorm is
a norm \cite{ivanovn}, \cite{soma}.


\subsection{Elementary properties of bounded cohomology}

As we will see later, bounded cohomology is not as tame as singular
cohomology because bounded cohomology fails to satisfy the excision
axiom (so the usual divide-and-conquer approach does not work). 
But at least bounded cohomology is natural, homotopy invariant and 
fulfils the dimension axiom.

\begin{lem}[Naturality]%
  \tindex{bounded cohomology!naturality of}%
  \tindex{naturality!of bounded cohomology}%
  \nindex{Hob}{$\bch * f$}{homomorphism induced by~$f$}% 
  Let $X,Y$ be topological spaces and $f \colon X \longrightarrow Y$ a
  continuous map. For each $k \in \N$, the map $f$ induces a
  homomorphism 
  \begin{align*} 
    \bch{k}{f} \colon \bch{k}{Y} & \longrightarrow \bch{k}{X}\\
                      [g]        & \longmapsto
                                   \bigl[ c \mapsto 
                                          g(f \circ c)\bigr]
  \end{align*}
  of operator norm at most 1. 
  Here composition of $f$ with elements of
  $\csingc{k}{X}{\R}$ is defined by linear extension of composition
  with elements of $\ssing{k}{X}.$
\end{lem}
\begin{bew}
  Let $g \in \bchn{k}{Y}$ with $\cobou(g)=0.$ Since $g$
  is bounded on $\ssing{k}{Y},$ the cochain  
  \begin{align*}
    \csingc{k}{X}{\R} & \longrightarrow \R \\
                    c & \longmapsto     g(f \circ c)
  \end{align*}
  is bounded on $\ssing{k}{X}$. According to the definition of the
  coboundary operator, this cochain is a cocycle. As
  the definition of $\bch{k}{f}([g])$ does not depend on the chosen
  representatives $g,$ the map $\bch{k}{f}$ is indeed
  well-defined. It is clear that $\bch{k}{f}$ is a homomorphism of
  real vector spaces and that
  \[ \supn[big]{\bch{k}{f}\bigl([g]\bigr)}
     \leq \supn{g}.\]
  Taking the infimum over all representatives yields
  \[\supn[big]{\bch{k}{f}\bigl([g]\bigr)} \leq \supn{[g]}.\qedhere\]
\end{bew}


\begin{lem}[Homotopy Invariance]\label{bchomotopyinv}%
  \tindex{bounded cohomology!homotopy invariance of}%
  \tindex{homotopy invariance!of bounded cohomology}
  The functor 
  bounded cohomology is homotopy invariant, i.e., if $f,g \colon X
  \longrightarrow Y$ are homotopic maps of topological spaces, then the
  induced maps are equal:
  \[ \bch{*}{f} = \bch{*}{g} \colon \bch{*}{Y} \longrightarrow \bch{*}{X}.\]
\end{lem}
\begin{bew}
  Since $f$ and $g$ are homotopic maps, there is a chain homotopy 
  \[ k_*
  \colon \csingc{*}{X}{\R} \longrightarrow \csingc{*+1}{Y}{\R} \]
  between $\csingc{*}{f}\R$ and $\csingc{*}{g}\R$ 
  \cite[Corollary~IV~16.5]{bredon}, i.e.,
  \[ \bou  \circ k_{*+1} + k_* \circ \bou 
     = \csingc{*+1}{f}\R - \csingc{*+1}{g}\R.\]
  This chain homotopy induces the map
  \begin{align*}
    h^* \colon \cosingc{*}{Y}{\R} & \longrightarrow \cosingc{*-1}{X}{\R}\\
                                c & \longmapsto 
                                   (-1)^{*} \cdot c \circ k_{*-1}.
  \end{align*}
  Obviously, $h^*$ maps bounded cochains to bounded cochains, thus it
  restricts to
  \[ \widehat h ^* := h^*|_{\bchn{*}{Y}} \colon \bchn{*}{Y} \longrightarrow
     \bchn{*-1}{X}.\]
  Since 
  \begin{align*}
    \bigl(\cobou \circ \widehat h^* + \widehat h^{*+1} \circ \cobou\bigr)(c)
      & = \cobou\bigl((-1)^* \cdot c \circ k_{*-1}\bigr) 
        + \widehat h^{*+1}\bigl((-1)^{*+1} \cdot c \circ \bou\bigr)\\
      & = c \circ k_{*-1} \circ \bou + c \circ \bou \circ k_* \\
      & = c \circ \bigl( \csingc{*}{f}\R - \csingc{*}{g}\R\bigr) \\
      & = \bigl(\bchn{*}{f} - \bchn{*}{g}\bigr)(c)
  \end{align*}
  holds for all $c \in \bchn{*}{Y},$ it follows that $\widehat h^*$ is a
  cochain homotopy between $\bchn{*}{f}$ and $\bchn{*}{g}.$ Hence we
  get $\bch{*}{f} = \bch{*}{g}.$
\end{bew}


\begin{lem}[Dimension Axiom]%
  \tindex{bounded cohomology!dimension axiom}%
  \tindex{dimension axiom!bounded cohomology}
  Bounded cohomology satisfies the dimension axiom, i.e.,
  \begin{align*}
                           \bch{0}{\opsp} & \cong \R, \text{ and}\\
    \fa{k \in\N_{\geq  1}} \bch{k}{\opsp} & = 0.
  \end{align*}  
\end{lem}
\begin{bew}
  This is the same calculation as in singular cohomology.
\end{bew}


\section{Duality}\label{dualitysection}
\tindex{duality}

In this section the duality between the seminorms $\lone{\,\cdot\,}$
and $\supn{\,\cdot\,}$ on singular homology and bounded cohomology 
respectively is explored. Using the
Hahn-Banach theorem we will see that the duality on the level of
(co)chain complexes carries over to (bounded co)homology. 
This duality was discovered and applied by Gromov \cite{gromov}.

The duality principle (Theorem~\ref{duality}) shows an important 
aspect of bounded cohomology: bounded
cohomology can be used to compute the simplicial volume. Since bounded
cohomology is much better understood (in view of the techniques
presented in Section~\ref{bcgroupssection}) 
than the seminorm on homology, duality leads to
interesting applications. For example, bounded cohomology can be used
to give estimates of the simplicial volume of products and connected sums
of manifolds.

Moreover, duality will play a central r\^ole in the proof that measure
homology and singular homology are \emph{isometrically} isomorphic
(see Section~\ref{isometricisom}). 

\begin{satz}[Duality Principle]\label{duality}
  \tindex{duality principle!of bounded cohomology}
  \tindex{bounded cohomology!duality principle of}
  Let $X$ be a topological space, $k\in \N$ and $\alpha \in
  \hsingc{k}{X}{\R}.$
  \begin{enumerate}
    \item Then $\lone{\alpha} = 0$ if and only if
          \[ \fa{\varphi \in \bch{k}{X}} \krp\varphi\alpha =0.\]
    \item If $\lone{\alpha} > 0,$ then
          \[ \frac{1}{\lone\alpha} 
             = \inf\bigl\{ \supn{\varphi} 
                   \bigm| \text{$\varphi\in\bch{k}{X}$,
                                $\krp\varphi\alpha = 1$}
                   \bigr\}.\]
  \end{enumerate}
\end{satz}

The theorem makes use of the Kronecker product on bounded cohomology.
Therefore, we recall the definition of the Kronecker product on  
singular (co)homology:

\begin{defi}
  Let $X$ be a topological space, $R$ a commutative ring with unit and
  $k \in \N.$ Furthermore, let $f \in \cosingc{k}{X}{R}$ be a cocycle and
  $a \in \csingc{k}{X}{R}$ a cycle. Then the \defin{Kronecker product}
  of $[f] \in \cohsingc{k}{X}{R}$ and $[a] \in \hsingc{k}{X}{R}$ is
  defined by 
  \dindex{Kronecker product}%
  \nindex{$d}{$\krp{\,\cdot\,}{\cdot\,}$}{Kronecker product}%
  \[ \krp[big]{[f]}{[a]} := f(a). \qedhere\] 
\end{defi}

In fact, the relations between the coboundary and the boundary
operator imply that the Kronecker product is well-defined.

\begin{defi}
  Let $X$ be a topological space. 
  The homomorphsim \[c_X \colon \bch{*}{X}
  \longrightarrow \cohsingc{*}{X}{\R}\] induced by the inclusion 
  $\bchn{*}X \subset
  \csingc{*}{X}{\R}$ is called
  \defin{comparison map}.%
  \dindex{comparison map}%
  \nindex{cX}{$c_X$}{comparison map}%
\end{defi}

In general, the comparison map is neither injective nor surjective:
for $S^1$ it is not injective, for $S^1\lor S^1$ it cannot be
surjective (Corollary~\ref{bcafg} and Example~\ref{bcexcex}).

\begin{defi}
  Let $X$ be a topological space, $k \in \N$, $\varphi \in
  \bch{k}{X}$, and $\alpha \in \hsingc{k}{X}{\R}.$ Then the \defin{Kronecker
  product} of $\varphi$ and $\alpha$ is defined to be 
  \dindex{Kronecker product!for bounded cohomology}
  \dindex{bounded cohomology!Kronecker product for}
  \nindex{$d}{$\krp{\,\cdot\,}{\cdot\,}$}{Kronecker product}
  \[ \krp\varphi\alpha := \krp[big]{c_X(\varphi)}{\alpha}.\qedhere\]
\end{defi}

\begin{bew}[Proof (of Theorem \ref{duality}).]
  Let $\alpha$ be represented by the cycle 
  \[ a=\fsum_{\sigma \in \ssing{k}{X}}
  a_\sigma \cdot \sigma \in \csingc{k}{X}{\R}.\]
  Assume that there is a $\varphi\in\bch{k}{X}$, represented by the
  bounded cochain $f \in \bchn{k}{X}$, with $\krp\varphi\alpha = 1.$
  Then, by definition, 
  \begin{align*}
    1 & =    |\negthinspace\krp\varphi\alpha | 
        =    \biggl|\, f\biggl(\fsum_{\sigma \in \ssing{k}{X}}
                             a_\sigma \cdot \sigma \biggr)
             \biggr|
        \leq \fsum_{\sigma\in\ssing{k}{X}} 
             |a_\sigma| \cdot |f(\sigma)| \\
      & \leq \supn{f} \cdot \lone{a}.
  \end{align*}
  If we take the infimum over all representatives of $\varphi$ and
  $\alpha,$ it follows that
  \[ 1 \leq \supn{\varphi} \cdot \lone{\alpha}.\]
  In particular, if $\lone{\alpha} = 0,$ there can be no such
  $\varphi.$ 

  Now, let $\lone{\alpha} > 0,$ and let $a \in \csingc{k}{X}{\R}$ be a
  cycle representing $\alpha.$ 
  We denote the closure in~$\csingc k X \R$ with respect to the norm
  $\lone{\;\cdot\;}$ by $\onec{\,\cdot\,}.$
  Thus the condition $\lone a \geq \lone \alpha > 0$ implies 
  \[ a \notin \onec{\im\bigl(\bou
     \colon \csingc{k+1}{X}{\R} \rightarrow \csingc{k}{X}{\R}\bigr)}.\] 
  Hence the Hahn-Banach theorem yields the existence of a continuous
  functional $f \colon \csingc{k}{X}{\R}
  \longrightarrow \R$ with
  \begin{align*}
                 f(a) & = 1,\\
                 f|_{\onec{\im\bigl(\bou
                            \colon \csingc{k+1}{X}{\R} 
                            \rightarrow \csingc{k}{X}{\R}\bigr)}}
                      & = 0, \\
        \supn[big]{f} & = \frac{1}{\lone{a}}.
  \end{align*}
  The second property implies that $f :=
  \in \cosingc{k}{X}{\R}$ is a
  cocycle. Moreover, $\supn{f} < \infty,$
  which shows $f \in \bchn{k}{X}.$ 
  By construction, we have  
  \[ \krp[big]{[f]}\alpha = \krp[big]{[f]}{[a]} = f(a) 
                          = 1\]
  and 
  \begin{align*}
    \supn{[f]} & \leq \supn{f} 
                 = \frac{1}{\lone{a}} 
                 \leq \frac{1}{\lone{\alpha}}.
  \end{align*}
  In particular, there exists a $\varphi \in \bch{k}{X}$ with
  $\krp\varphi\alpha = 1$ and $\supn \varphi \leq 1/\lone\alpha$.
  Hence, if $\lone{\alpha} >0,$
  \[ \frac{1}{\lone\alpha} 
             = \inf\bigl\{ \supn{\varphi} 
                   \bigm| \text{$\varphi\in\bch{k}{X}$,
                                $\krp\varphi\alpha = 1$}
                   \bigr\}.\qedhere\]
\end{bew}

\begin{koro}\label{loneisom}\tindex{isometric isomorphism}
  Let $f \colon X \longrightarrow Y$ be a continuous map such that the
  induced map 
  $\bch * f \colon \bch * Y \longrightarrow \bch * X$ is an isometric
  isomorphism. Then
  \[ \lone\alpha = \lone[big]{\hsingc * f \R (\alpha)}\]
  for all $\alpha \in \hsingc * X \R$.
\end{koro}
\begin{bew}
 Since $\bch * f$ is an isometric isomorphism, we obtain 
 \begin{align*}
      & \bigl\{ \supn{\varphi}
        \bigm| \text{%
               $\varphi \in \bch{n}{Y}$, 
               $\krp[big]\varphi{\hsingc{n}{f}{\R}(\alpha)} =1$}
       \bigr\} \\
    & \qquad =  \bigl\{ \supn[big]{\bch{n}{f}(\varphi)}
        \bigm| \text{%
               $\varphi \in \bch{n}{Y}$, 
               $\krp[big]{\bch{n}{f}(\varphi)}{\alpha} =1$}
       \bigr\} \\
    & \qquad =  \bigl\{ \supn{\psi}
        \bigm| \text{%
               $\psi \in \bch{n}{X}$, 
               $\krp\psi{\alpha} =1$}
       \bigr\}.
  \end{align*}
  Therefore application of Theorem~\ref{duality} proves the Corollary.
\end{bew}

\subsection{Bounded cohomology and simplicial volume}%
\tindex{simplicial volume!bounded cohomology and}%
\tindex{bounded cohomology!simplicial volume and}

Application of the duality principle to the fundamental class yields a
description of the simplicial volume in terms of bounded cohomology: 

\begin{koro}\label{dualm}
  Let $M$ be an oriented closed connected $n$-dimensional manifold.
  \begin{enumerate}
    \item Then $\sv{M}=0$ if and only if the comparison homomorphism
          \[ c_M^n \colon \bch{n}{M} 
                        \longrightarrow \cohsingc{n}{M}{\R}\]
          is trivial.
    \item If $\sv{M} > 0,$ then
          \[ \frac{1}{\sv{M}} = \supn[big]{\cfcl{M}},\]
          where $\cfcl{M} \in \cohsingc{n}{M}\R$ denotes the
          image of the cohomological fundamental class of $M$ 
          under the change of rings homomorphism. 
  \end{enumerate}
\end{koro}
\begin{bew}
By definition, $\krp[big]{\cfcl{M}}{\fclr{M}} = 1$. Furthermore,
\[ \cohsingc{n}{M}{\R} \cong \R \cdot \cfcl{M}.\]
Hence $\im(c_M^n) \neq 0$ if and only if $\cfcl{M} \in \im(c_M^n).$

1. Therefore the first part of Theorem~\ref{duality} yields that $\sv{M} =
 \lone{\fclr{M}} = 0$ if and only if $c_M^n$ is trivial.

2. Suppose $\sv{M} > 0.$ By the first part, 
   $c_M^n \colon \bch{n}{M} \longrightarrow \cohsingc{n}{M}{\R} \cong
   \R$ is not trivial and hence surjective. Moreover, 
   \begin{align*}
        \supn{\varphi} & = \inf \bigl\{
                             \supn{f}
                           \bigm|\text{%
                              $f \in \bchn{n}{X}$, $\cobou(f) = 0$,
                              $[f] = \varphi$ in $\bch{n}{M}$}
                           \bigr\} \\
                       & \geq \inf \bigl\{
                             \supn{f}
                           \bigm|\text{%
                              $f \in \cosingc{n}{X}{\R}$, 
                              $\cobou(f) = 0$, 
                              $[f] = c_M^n(\varphi)$ in $\cohsingc{n}{M}{\R}$}
                           \bigr\} \\
                       & = \supn{c_M^n(\varphi)}
   \end{align*}
   holds for all $\varphi \in \bch{n}{M}$. 
   Using the surjectivity of $c_M^n$, we obtain (by the second part of
   Theorem~\ref{duality})
   \begin{align*}
       \frac1{\sv{M}} & = \frac1{\lone[big]{\fclr{M}}} \\
                      & = \inf \bigl\{
                              \supn{\varphi}
                           \bigm|\text{%
                              $\varphi \in \bch nM$, 
                              $\krp\varphi{\fclr M} = 1$}
                           \bigr\} \\
                      & \geq \inf \bigl\{
                              \supn{c_M^n(\varphi)}
                           \bigm|\text{%
                              $\varphi \in \bch nM$, 
                              $\krp[big]{c_M^n(\varphi)}{\fclr M} = 1$}
                           \bigr\} \\
                      & \geq \inf \bigl\{
                              \supn{\psi}
                           \bigm|\text{%
                              $\psi \in \cohsingc{n}{M}{\R} 
                                        \cong \cfcl{M} \cdot \R$, 
                              $\krp\psi{\fclr M} = 1$}
                           \bigr\} \\
                      & = \supn[big]{\cfcl M}.
   \end{align*}
   In particular, $\supn[big]{\cfcl M}$ is finite. On the other hand,  
   for each $\varepsilon \in\R_{>0}$ there is a cocycle $f \in
   \cosingc{n}{M}{\R}$ with $[f] = \cfcl M$ in $\cohsingc{n}{M}{\R}$
   and
   \[ \supn f \leq \supn[big]{\cfcl M} + \varepsilon < \infty.\]
   Thus $f \in \bchn{n}{M}.$ Let $\varphi \in \bch{n}{M}$ be the
   corresponding bounded cohomology class. Then 
   \[ \supn{\varphi} \leq \supn f \leq \supn[big]{\cfcl{M}} + \varepsilon\]
   and
   \[ \krp\varphi{\fclr M} = \krp{c_M^n(\varphi)}{\fclr M}
                           = \krp{[f]}{\fclr M}
                           = \krp[big]{\cfcl M}{\fclr M}
                           = 1, \]
   which implies
   \[ \frac1{\sv{M}} =  \frac1{\lone[big]{\fclr M}}
                     \leq \supn{\varphi}
                     \leq \supn[big]{\cfcl M} + \varepsilon. \qedhere\]
\end{bew}

In Subsection~\ref{apptosv} we will combine this result with the
mapping theorem (Theorem~\ref{mt}) to compute the simplicial volume in some
special cases. 

\subsection{The first bounded cohomology group}
\tindex{bounded cohomology!first group}

Using the result of Theorem~\ref{firsthom} it is not hard to see that the
first bounded cohomology group always vanishes:

\begin{koro}
  For all topological spaces $X$ the group~$\bch 1 X$ is zero.
\end{koro}
\begin{bew}
  By Theorem~\ref{firsthom} the seminorm on~$\hsingc 1 X \R$ is trivial
  if $X$ is arcwise connected. Since the singular homology of a space
  is the direct sum of the singular homology groups of its path
  components, this result carries over to all spaces~$X$.

  Now let $\varphi \in \bch{1}{X}$ and let $f \in \bchn{1}{X}$ be a cocycle
  representing $\varphi.$ Since the seminorm on $\hsingc 1 X \R$ is
  trivial (Theorem~\ref{firsthom}), we get from Theorem~\ref{duality} that
  \[ \fa{\alpha \in \hsingc{1}{X}{\R}} 0 = \krp\varphi\alpha.\]
  By the universal coefficient theorem,
  \begin{align*} 
   \cohsingc{1}{X}{\R} & \longrightarrow \hom\bigl(\hsing{1}{X},\R\bigr)\\
                  \psi & \longmapsto     \bigl(\alpha \mapsto \krp\psi\alpha \bigr)
  \end{align*}
  is an isomorphism.
  Hence the comparison map $c_X^1 \colon \bch{1}{X} \longrightarrow
  \cohsingc{1}{X}{\R}$ must be trivial. Thus there is a cochain $u \in
  \cosingc0X\R$ with
  \[ \cobou(u) = f.\]
  However, in general $u$ is not bounded. Altering $u$, we construct
  a bounded cochain~$\widehat u \in \bchn{0}{X}$ satisfying 
  $\cobou(\widehat u) = f:$
  Let $(X)_{i \in I}$ be the path components of $X.$ For each $i \in
  I$ we choose a point $x_i \in X_i.$ If $x \in X$,  then there is
  exactly one $i(x) \in I$ such that $x \in X_{i(x)}.$ We define
  $\widehat u \in \cosingc{0}{X}\R$ to be the linear extension of
  \[ \fa{x \in X} \widehat u (x) := u(x) - u(x_{i(x)}).\]
  Since $x$ and $x_{i(x)}$ are in the same path component, there is a
  path $\sigma_x$ in $X$ joining these points. So
  \begin{align*}
    \bigl|\widehat u(x)\bigr| 
      & = \bigl| u(\sigma_x(0)) - u(\sigma_x(1)) \bigr| \\
      & = \bigl| u(\bou(\sigma_x))\bigr| \\
      & = \bigl| \cobou(u)(\sigma_x)\bigr| \\
      & = \bigl| f(\sigma_x)\bigr| \\
      & \leq \supn{f}, 
  \end{align*}
  which implies $\widehat u \in \bchn{0}{X}.$ Furthermore we have for
  all $\sigma \in \ssing1X$
  \begin{align*}
    \cobou(\widehat u)(\sigma) 
      & = \widehat u(\sigma(1) - \sigma(0)) \\
      & = \widehat u(\sigma(1)) - \widehat u(\sigma(0)) \\
      & = u(\sigma(1)) - u(x_{i(\sigma(1))})
        - u(\sigma(0)) + u(x_{i(\sigma(0))}).
  \end{align*}
  Here $i(\sigma(1)) = i(\sigma(0))$ because $\sigma(1)$ and
  $\sigma(0)$ lie in the same path component. Hence
  \begin{align*}
    \cobou(\widehat u)(\sigma)
      & = u(\sigma(1)) - u(\sigma(0)) \\
      & = u(-\bou(\sigma)) \\
      & = \cobou(u)(\sigma) \\
      & = f(\sigma).
  \end{align*}
  This proves $\varphi = [f] = 0$ in $\bch{*}{X}.$
\end{bew}


\subsection{Products of manifolds}%
\tindex{simplicial volume!of products}

Bounded cohomology makes it possible to prove that the simplicial
volume is compatible with taking products (up to a constant) \cite[page~218]{gromov}.

\begin{satz}\label{prod}
  For each $n \in \N_{>0}$ there is a constant $c(n) \in \R_{>0}$ such
  that the following holds: If $M$ and $N$ are two oriented closed
  connected manifolds, then
  \[      \sv N \cdot \sv M 
     \leq \sv{N \times M} 
     \leq c(\dim N + \dim M) \cdot \sv N \cdot \sv M. \]
\end{satz}

The proof makes use of explicit descriptions of the cup-product and the
cross-products in singular (co)homology. In order to see the first
inequality, duality is applied. A detailed proof is given in the book
of Benedetti and Petronio \cite[Theorem~F.2.5]{bp}.  The
constant~$c(n)$ is related to the number of $n$-simplices needed
to triangulate the products~$\ssim p \times \ssim {n-p}$ for all $p
\in \{1, \dots, n\}$.

% -------------------------------------------------------------
\section{Bounded cohomology of groups}\label{bcgroupssection}
\tindex{bounded cohomology!of groups}

In his paper ``Foundations of the Theory of Bounded Cohomology,''
Ivanov developed a transparent derived functor type approach to bounded cohomology
\cite{ivanov}. Just like Ivanov, we will consider bounded cohomology
of discrete groups, but there is also a theory available for
topological groups \cite{monod}.

\subsection{Relatively injective resolutions}

Similar to ordinary group cohomology, the bounded cohomology of a
group is based upon special kinds of resolutions. The basic objects
are relatively injective bounded $G$-modules. 

\begin{defi}
  Let $G$ be a discrete group.
  \begin{itemize}
   \item \dindex{bounded Gmodule@bounded $G$-module}%
         \dindex{Gmodule@$G$-module!bounded}%
         A \defin{bounded} $G$\defin{-module} is a Banach space~$V$ 
         with a (left) $G$-action such that $\| g\cdot
         v\| \leq \| v \|$ for all $g \in G$ and all $v \in V$.
   \item \dindex{Gmorphism@$G$-morphism}
         A $G$\defin{-morphism} of bounded $G$-modules is a
         bounded $G$-equivariant linear operator.
   \item \dindex{strongly injective}
         \dindex{Gmorphism@$G$-morphism!strongly injective}
         An injective $G$-morphism $i \colon V \longrightarrow W$
         between bounded $G$-modules is said to be \defin{strongly
         injective} if there is a bounded linear operator $s \colon W
         \longrightarrow V$ satisfying $s \circ i = \id$ and $\|s\|
         \leq 1$ (but $s$ does not need to be $G$-equivariant).
   \item \dindex{relatively injective!Gmodule@$G$-module}
         \dindex{Gmodule@$G$-module!relatively injective}
         A bounded $G$-module~$U$ is \defin{relatively injective} if for
         any strongly injective $G$-morphism $i \colon V
         \longrightarrow W$ 
         and any $G$-morphism $\alpha\colon V \longrightarrow U$
         of bounded $G$-modules there is a 
         $G$-morphism $\beta \colon W \longrightarrow U$ of bounded
         $G$-modules such that $\beta \circ i = \alpha$ and $\|\beta\|
         \leq \|\alpha\|$. \qedhere
  \end{itemize}
\end{defi}

Good candidates for relatively injective bounded $G$-modules are
mapping spaces:

\begin{defi}
  \nindex{BG}{$B(G,V)$}{set of bounded functions from~$G$ to~$V$\\}
  Let $G$ be a discrete group and let $V$ be a Banach space.  
  We 
  write~$B(G,V)$ for the real vector space of bounded functions~$G
  \longrightarrow V$. This vector space admits a left $G$-action via
  \dindex{group action!on $B(G,V)$}
  \[ \fa{f \in B(G,V)} \fa{g \in G} g\cdot f := 
                                   \bigl( h \mapsto f(hg)\bigr). 
     \qedhere \]
\end{defi}

\begin{bsp}
  Let $V$ be a Banach space and let $G$ be a discrete group. The
  supremum norm turns $B(G,V)$ into a bounded $G$-module. Moreover,
  this bounded $G$-module is relatively injective \cite[Lemma
  (3.2.2)]{ivanov}. 
\end{bsp}

\begin{defi}
  \dindex{Gresolution@$G$-resolution}
  Let $G$ be a discrete group. A $G$\defin{-resolution} of a bounded
  $G$-module~$V$ is a resolution of~$V$ via bounded $G$-modules and
  $G$-morphisms
  \[ \xymatrix@=2em{
       {0} \ar[r] &  {V} \ar[r] & {V_0} \ar[r] & {V_1} \ar[r] & {\cdots}  
     }
  \]
  \dindex{strong}%
  \dindex{Gresolution@$G$-resolution!strong}%
  Such a resolution is \defin{strong} if there is a
  \dindex{contracting homotopy}%
  \defin{contracting homotopy}, i.e., there is a chain
  contraction~$(K_n)_{n \in \N}$ consisting of (not necessarily
  equivariant) linear operators with
  $\|K_n\| \leq 1$ for all~$n \in \N$. 
  \dindex{relatively injective!Gresolution@$G$-resolution}
  \dindex{Gresolution@$G$-resolution!relatively injective}
  The above resolution is called
  \defin{relatively injective} if the bounded $G$-modules~$(V_n)_{n
  \in \N}$ are relatively injective. 
\end{defi}

\begin{bsp}\label{stdres}
  Let $G$ be a discrete group. There always exists a strong relatively
  injective resolution of the trivial $G$-module~$\R$, the
  \defin{(inhomogeneous) standard resolution}:\dindex{standard resolution}%
  \dindex{Gresolution@$G$-resolution!standard}%
  
  For $n \in \N$ we define $V_n := B(G^n,\R)$. Equipped with the
  supremum norm and the left
  $G$-action\dindex{group action!on~$B(G^n,\R)$} given by
  \begin{align*} 
    g\cdot f := \bigl( (g_1, \dots, g_n) \mapsto 
                       f(g_1, \dots, g_{n-1}, g_n\cdot g) 
                \bigr)
  \end{align*}
  the module~$V_n$ becomes a relatively injective bounded
  $G$-module. The homomorphisms
  \begin{align*}
    d_{-1} \colon \R & \longrightarrow V_0 \\
                   c & \longmapsto     (g \mapsto c)
  \end{align*}
  and (for $n\in \N$)
  \begin{alignat*}{2}
    d_n \colon V_n & \longrightarrow V_{n+1} && \\
                 f & \longmapsto
        \Bigl( (g_0, \dots, g_{n+1}) \mapsto && 
                (-1)^{n+1} \cdot f(g_1, \dots, g_{n+1}) \\
     &  &&      + \sum_{j=1}^{n} (-1)^{n-j} \cdot f(g_0, \dots, 
                                                  g_jg_{j+1}, \dots, 
                                                  g_{n+1})
        \Bigr)
  \end{alignat*}
  turn $(V_n)_{n\in\N}$ into a strong relatively injective resolution
  \cite[(3.4)]{ivanov}. Analogously to the classical case, one can also
  define the corresponding homogeneous resolution. 
\end{bsp}

There is also a version of the fundamental theorem of homological
algebra in the setting of bounded $G$-modules \cite[Lemma~(3.3.2)]{ivanov},
\cite[Lemma 7.2.4~and~7.2.6]{monod}:

\begin{satz}\label{fundth}
  \tindex{fundamental theorem of homological algebra}
  \tindex{Gresolution@$G$-resolution}
  Let $G$ be a discrete group and let $U$ and $V$ be bounded
  $G$-modules. Suppose 
  \[ \xymatrix@=2em{
        {0} \ar[r]
      & {V} \ar[r]^{\varepsilon_V}
      & {V_0} \ar[r]^{v^0}
      & {V_1} \ar[r]^{v^1}
      & {\cdots}
     }
  \]
  is a strong resolution of~$V$ and 
  \[ \xymatrix@=2em{
        {0} \ar[r]
      & {U} \ar[r]^{\varepsilon_U}
      & {U_0} \ar[r]^{u^0}
      & {U_1} \ar[r]^{u^1}
      & {\cdots}
     }
  \]
  is a complex of relatively injective bounded $G$-modules. Then any
  $G$-morphism $w \colon V \longrightarrow U$ can be extended to a
  $G$-morphism $(w_n)_{n\in\N}$ of complexes:
  \begin{align*}
   \xymatrix@=2em{
        {0}   \ar[r]
      & {V}   \ar[r]^{\varepsilon_V} \ar[d]^w
      & {V_0} \ar[r]^{v^0}           \ar[d]^{w_0}
      & {V_1} \ar[r]^{v^1}           \ar[d]^{w_1}
      & {\cdots} \\
      % -------------------
        {0}   \ar[r]
      & {U}   \ar[r]_{\varepsilon_U}
      & {U_0} \ar[r]_{u^0}
      & {U_1} \ar[r]_{u^1}
      & {\cdots.}
   }
  \end{align*}
  Any two such extensions are chain homotopic via a $G$-equivariant
  homotopy. 
\end{satz}

\subsection{Definition of bounded cohomology of groups}

In view of Theorem~\ref{fundth}, bounded cohomology of groups can
be defined in the same way as usual group cohomology:

\begin{defi}
  \nindex{VG}{$V^G$}{$G$-fixed points in~$V$}
  If $G$ is a discrete group and $V$ is a bounded $G$-module, we write
  $V^G$ for the set of $G$-fixed points. 
\end{defi}

\begin{defi}
  Let $G$ be a discrete group and let 
  \[ \xymatrix@=2em{{\R} \ar[r] 
   & {V_0} \ar[r]^{v^0} 
   & {V_1} \ar[r]^{v^1}
   & {\cdots}}\]
  be a strong relatively injective resolution of the trivial bounded
  $G$-module~$\R$. 
  \dindex{bounded cohomology!of groups}%
  \nindex{Hob}{$\bch * X$}{bounded cohomology of~$X$}%
  Then the \defin{bounded cohomology of}~$G$ (with
  coefficients in~$\R$), $\bch * G$, is the cohomology of the induced complex
  \[ \xymatrix@=2em{
       {0} \ar[r]
     & {V_0^G} \ar[r]^{v^0|}
     & {V_1^G} \ar[r]^{v^1|}
     & {\cdots.}
     }
     \qedhere
  \]
\end{defi}

Of course, one could also consider bounded cohomology of groups with
more general coefficients \cite{monod}. But in the following we will 
need only bounded cohomology with real coefficients.

\begin{koro}\label{bcgcor}
  \tindex{Gresolution@$G$-resolution}
  Let $G$ be a discrete group. Then any $G$-morphism between two strong
  relatively injective resolutions of the trivial bounded 
  $G$-module~$\R$ by bounded
  $G$-modules extending the identity $\R \longrightarrow \R$ induces
  the same isomorphism on the cohomology of the $G$-fixed point sets. 
  
  In particular, the bounded cohomology~$\bch * G$ is well-defined (up
  to canonical isomorphisms). 
\end{koro}

\begin{bew}
  This follows directly from Theorem~\ref{fundth}. A detailed proof is
  given by Ivanov \cite[Section~(3.7)]{ivanov}. 
\end{bew}

Each strong relatively injective resolution of the trivial
$G$-module~$\R$ gives rise to a seminorm on~$\bch * G$. The topology
on~$\bch * G$ is independent of the chosen resolution, but the seminorm
does depend on the resolution. However, it is known that the standard
resolution (compare Example~\ref{stdres}) induces the minimum of all
these seminorms \cite[Theorem~(3.6)]{ivanov}, the so-called
\defin{canonical seminorm}.%
\dindex{canonical seminorm}%
\dindex{seminorm!canonical}


\nindex{Hob}{$\bch * f$}{homomorphism induced by~$f$}% 
As in the case of ordinary group cohomology, each group homomorphism $f
\colon G \longrightarrow H$ induces a homomorphism $\bch * f \colon
\bch * H \longrightarrow \bch *G$ (for a rigorous definition
see~\cite[Section~(3.7)]{ivanov}). This turns bounded cohomology of
groups into a functor.

As one might suspect, $\bch * G \cong \bch * {BG}$ if $BG$ is a model
of the classifying space of~$G$. In fact, much more holds, as is
explained in Section~\ref{mtsec}.

\subsection{Amenability}\label{amenability}

An interesting facet of bounded cohomology is that it ignores a large
class of groups -- the amenable groups. This gives rise to interesting
applications to the simplicial volume (see Subsection~\ref{apptosv}).

\begin{defi}
  \dindex{amenable}%
  A group $G$ is called \defin{amenable} if the there is a
  $G$-invariant function $m \colon B(G,\R) \longrightarrow \R$ such
  that for all $f \in B(G,\R)$
  \begin{align*}
    \inf_{g \in G} f(g) & \leq m(f)  
                          \leq \sup_{g \in G} f(g). \qedhere
  \end{align*}
\end{defi}

Actually, this might seem a far-fetched definition. But amenable
groups turn out to be quite interesting -- for example they are
connected with the Banach-Tarski paradox \cite[Chapter~0]{runde}. 

In the above definition, $m$ should be interpreted as a kind of
special mean on~$G$. Obviously, all finite groups are amenable -- in
this case $m$ can be taken as integration over~$G$ with respect to the
counting measure and dividing by the number of elements in~$G$.  
Furthermore, all Abelian groups are amenable \cite[Example~1.1.5]{runde}.

Ivanov shows that bounded cohomology cannot detect amenable groups
\cite[Section~3.8]{ivanov}:

\begin{satz}\label{bcamenable}%
  \tindex{amenable}%
  \tindex{isometric isomorphism}
  Let $A$ be a normal amenable subgroup of the discrete
  group~$G$. Then the map $\bch * {G/A} \longrightarrow \bch * G$
  induced by the canonical homomorphism $G \longrightarrow G/A$ is an
  isometric isomorphism.
\end{satz}

Using the right invariant ``mean'' on~$A$ Ivanov proves that the
standard resolution of the quotient group~$G/A$ also is strong
relatively injective resolution of~$\R$ by bounded
$G$-modules. Moreover, the induced canonical isomorphism turns out to
be isometric. 

The theorem shows that the free group $\Z * \Z$ cannot be amenable: It
can be shown that $\bch 2 {\Z * \Z}$ is not even finitely
generated. The ``proof'' of Brooks \cite{brooks} gives good intuition
about this fact, but is not entirely correct -- a correct version can
be found in the arcticles of Grigorchuk and Mitsumatsu \cite{grigo},
\cite{matsumoto}.



% ------------------------------------------------------------
\section{The mapping theorem}\label{mtsec}

Perhaps surprisingly, the rather small differences between the
definitions of singular and bounded cohomology lead to significant
differences in the behaviour of both theories. For instance, 
Gromov and Ivanov showed that the  
bounded cohomology of spaces only 
depends on the fundamental group (Theorem~\ref{bcandfg}). 
More precisely, if $f \colon X \longrightarrow Y$ is a continuous map,
there is a commutative diagram
\begin{align*}
  \xymatrix@=3.5em{%
    {\bch * Y} \ar[r]^{\bch * f} \ar[d]_{\cong} &
    {\bch * X} \ar[d]^{\cong} \\
    {\bch * {\pi_1(Y)}} \ar[r]_{\bch *{\pi_1(f)}} &
    {\bch * {\pi_1(X)}}
  }
\end{align*}
whose vertical arrows are canonical isometric isomorphisms. This
diagram allows us to understand the bounded cohomology of spaces via
Ivanov's transparent axiomatic approach of Section~\ref{bcgroupssection}.

\subsection{Bounded cohomology and the fundamental group}

One proceeds in two steps to obtain the fact that bounded cohomology
of a space only depends on its fundamental group: 

\begin{satz}
  \tindex{isometric isomorphism}
  Let $X$ be a simply connected \cw-complex with countably many
  cells. Then the bounded cohomology of~$X$ is canonically
  isometrically isomorphic to the bounded cohomology of the one point
  space. 
\end{satz}

\begin{satz}\label{bcandfg}%
  \tindex{isometric isomorphism}%
  \tindex{bounded cohomology!fundamental group and}
  Let $X$ be a connected countable \cw-complex. Then $\bch * X$ is canonically
  isometrically isomorphic to~$\bch[big] * {\pi_1(X)}$.
\end{satz}

Actually, these theorems remain true if we drop the condition that the
spaces must be countable cellular. This situation is treated
in Gromov's original exposition \cite[Section~3.1]{gromov}. 

Quite understandable proofs of the above theorems can be found
in the paper of Ivanov \cite{ivanov}. We will sketch these proofs: 
in the following, we view~$\bchn k X$ as the set of bounded
functions~$\ssing k X \longrightarrow \R$. 

To prove the first theorem a contracting chain homotopy
\begin{equation*}
  \xymatrix@=2em{%
    {\R} & {\bchn{0}{X}} \ar[l]_--{K_0}%
         & {\bchn{1}{X}} \ar[l]_--{K_1}%
         & {\cdots}      \ar[l]_--{K_2}%
  }
\end{equation*}
with $ \| K_n \| \leq 1$ for all $n \in \N$ is constructed explicitly 
using ``towers of connective covers.'' 
Here it is needed that $X$ is a countable
\cw-complex to ensure that its homotopy groups are all countable
\cite[Theorem~IV~6.1]{lundell}. 
This implies that there is a model of the Eilenberg-Mac~Lane 
space~$K\bigl(\pi_n(X),n \bigr)$ which is a topological Abelian
group~$G_n$.
\margcomm{na und??}
Then there is such a tower
\begin{equation*}
  \xymatrix@=2em{%
    {\cdots} \ar[r]^{p_3} 
  & {X_3}    \ar[r]^{p_2}
  & {X_2}    \ar[r]^{p_1}
  & {X_1=X} 
  }
\end{equation*}
consisting of principal $G_n$-bundles $p_n \colon X_{n+1}
\longrightarrow X_n$. Ivanov shows that in this situation for each~$n
\in \N$ there is a chain map~$A_n \colon \bchn * {X_{n+1}}
\longrightarrow \bchn * {X_n}$ satisfying 
\[ A_n \circ \bch * {p_n} = \id \quad \text{and} \quad 
   \|A_n\| = 1. \]
Using these $(A_n)_{n \in \N}$, the contracting homotopy~$K_*$ can
inductively be constructed. 

In order to prove the second theorem, the first theorem is applied to show 
that (where $p \colon \ucov{X} \longrightarrow X$ is the
universal covering of~$X$)
\begin{equation*}
  \xymatrix@=2em{%
    {0}                      \ar[r] &
    {\R}                     \ar[r] &
    {\bchn[norm]{0}{\ucov X}} \ar[r] &
    {\bchn[norm]{1}{\ucov X}} \ar[r] &
    \cdots }
\end{equation*}
is a strong relatively injective resolution of the trivial bounded
$\pi_1(X)$-mod\-ule~$\R$.
Here the left $\pi_1(X)$-action on $\bchn[norm] * {\ucov X}$ is given
by the right $\pi_1(X)$-action on the universal covering space~$\ucov
X$: for all $f \in \bchn k X$ and all $g \in \pi_1(X)$
\[ g \cdot f := \bigl( \sigma \longmapsto
                       f(x \mapsto \sigma(x) \cdot g)
                \bigr).
\]

Thus the bounded cohomology of the discrete group $\pi_1(X)$ can be 
computed as the cohomology of the complex 
\begin{equation*}
  \xymatrix@=2em{%
    {0}                                  \ar[r] &
    {\bchn[norm]{0}{\ucov{X}}^{\pi_1(X)}} \ar[r] &
    {\bchn[norm]{1}{\ucov{X}}^{\pi_1(X)}} \ar[r] &
    {\cdots}
  }
\end{equation*} 
The map $\bchn{*}{X} \longrightarrow\bchn[big]{*}{\ucov{X}}$ induced by~$p$ 
establishes an isomorphism between $\bchn * X$ and
$\bchn[norm]{*}{\ucov X}^{\pi_1(X)}$ commuting with the differentials. Hence the
bounded cohomology of $\pi_1(X)$ coincides with the cohomology of the
complex 
\begin{equation*}
  \xymatrix@=2em{%
    {0}           \ar[r]          &
    {\bchn{0}{X}} \ar[r]^--{\bou} &
    {\bchn{1}{X}} \ar[r]^--{\bou} &
    {\cdots}}
\end{equation*}
which is the bounded cohomology of the space $X$. 
A more detailed calculation reveals that the induced isomorphism
is actually isometric.

\begin{qedkoro}\label{bcafg}
  If $X$ is a connected countable \cw-complex with amenable
  fundamental group, then $\bch * X \cong \bch * \opsp$.
\end{qedkoro}

\begin{bsp}\label{bcexcex}%
\tindex{bounded cohomology!excision}%
\tindex{excision!bounded cohomology and}
The above relation between bounded cohomology and fundamental groups 
also yields that bounded cohomology cannot satisfy
excision and that we cannot expect to find a cellular version of
bounded cohomology: We have $\pi_1(S^1 \lor S^1) = \Z * \Z$, 
and $\bch 2 {\Z * \Z}$ is non-trivial (see
Subsection~\ref{amenability}), but $\bch * {S^1} \cong \bch * \opsp$. 
\end{bsp}

\begin{bsp}%
  \tindex{characteristic classes}%
  \tindex{bounded cohomology!characteristic classes and}
  There is no interesting theory of characteristic classes 
  for real vector bundles in bounded cohomology: 
  For each $n \in \N$, the fundamental group of the classifying
  space~$BO(n)$ is amenable. 
  Therefore the bounded cohomology~$\bch * {BO(n)}$ coincides with the
  bounded cohomology of the one point space.  
 
  However, there are interesting results about the supremum norm of
  the ordinary Euler class of orientable flat vector bundles 
  \cite[page~231]{gromov}. 
\end{bsp}

\subsection{The mapping theorem}

If $f \colon X \longrightarrow Y$ is a continuous map, chasing through
all the definitions and isomorphisms shows that $\bch * f =
\bch * {\pi_1(f)}$ holds with respect to the isometric isomorphisms
$\bch * X \cong \bch * {\pi_1(X)}$ and $\bch * Y \cong \bch *
{\pi_1(Y)}$ of Theorem~\ref{bcandfg}. Thus combining 
Theorem~\ref{bcandfg} and Theorem~\ref{bcamenable} yields: 

\begin{satz}[Mapping Theorem for Bounded Cohomology]\label{mt}%
  \tindex{mapping theorem}%
  \tindex{isometric isomorphism}
  Let $X$ and $Y$ be countable \cw-complexes and let $f \colon X
  \longrightarrow Y$ be a continuous map. If the homomorphism
  $\pi_1(f) \colon \pi_1(X) \longrightarrow \pi_1(Y)$ is surjective
  and its kernel is amenable, then the induced homomorphism
  \[ \bch * f = \bch * {\pi_1(f)} \colon 
     \bch * Y \longrightarrow \bch * X \]
  is an isometric isomorphism. 
\end{satz}

By Corollary~\ref{loneisom}, in the situation of Theorem~\ref{mt}, the
homomorphism $\hsingc * f \R$ then also preserves the seminorm in
homology.

Again, the mapping theorem remains true if we drop the condition for
the spaces to be countable cellular \cite[Section~3.1]{gromov}.

\subsection{Applications to simplicial volume}\label{apptosv}

In particular, we can apply the mapping theorem to obtain the
following results about the simplicial volume:

\begin{koro}\label{svbg}%
  \tindex{simplicial volume!classifying maps and}%
  \tindex{simplicial volume!amenability and}%
  \tindex{amenable}
  Let $M$ be an oriented closed connected manifold of positive
  dimension~$n$, and let $G$ be its fundamental group.
  \begin{enumerate}
    \item If $f \colon M \longrightarrow BG$ denotes the classifying 
          map, then
          \[ \sv M =  \lone{\hsingc n f \R(\fclr M)}.\]
    \item Moreover, if $G$ is amenable, then $\sv M = 0$.
  \end{enumerate}
\end{koro}
\begin{bew}
  1. By construction, $\pi_1(f)$ is an isomorphism. Hence the mapping
  theorem yields that $\bch n f$ is an isometric isomorphism. Now
  Corollary~\ref{loneisom} shows $\sv M =  \lone{\hsingc n f \R(\fclr M)}$.

  2. Corollary~\ref{bcafg} 
  implies that $\bch n M$ vanishes. Hence 
  \[\sv M = \lone[big]{\fclr M} = 0,\] 
  by duality (the first part of Theorem~\ref{duality}).
\end{bew}

A nice argument from covering theory yields the following result 
\cite[Exercise~14.11]{lueck}:

\begin{koro}%
  \tindex{simplicial volume!amenability and}%
  \tindex{amenable}
  Let $M$ and $N$ be oriented closed connected manifolds of the same 
  dimension and let $f \colon M
  \longrightarrow N$ be a continuous map of degree 1. Then $\pi_1(f)$
  is surjective. If the kernel of $\pi_1(f)$ is amenable, then
  \[ \sv{M} = \sv{N}. \]
\end{koro}

\begin{bsp}
  \tindex{simplicial volume!fudamental group and}
  However, the simplicial volume does not only depend on the
  fundamental group of the manifold: Let $F_2$ be the oriented closed
  connected surface of genus~$2$. Then $F_2$ and $F_2 \times S^2$ have
  the same fundamental group, but
  \[ \sv{F_2} = 4 \quad\text{and}\quad 
     \sv[norm]{F_2 \times S^2} = 0\]
  by Example~\ref{exfg} and Theorem~\ref{prod} respectively.
\end{bsp}

Furthermore, one can derive from the mapping theorem 
that the simplicial volume is
additive with respect to connected sums.

\begin{satz}\label{consum}%
  \tindex{simplicial volume!of connected sums}%
  Let $M$ and $N$ be two oriented closed connected manifolds of the
  same dimension~$n \geq 3$. Then
  \[ \sv{M \consum N} = \sv M + \sv N.\] 
\end{satz}

Gromov's proof of this theorem combines beautiful geometric ideas of
simplicial ``tree-like'' complexes with the mapping theorem (applied
to the evident map $M \consum N \longrightarrow M \lor N$). However,
his exposition in ``Volume and Bounded Cohomology'' 
\cite[Section~3.5]{gromov} is rather short.
% and lacks some details... :)

% -------------------------------------------------------------
\section{A special resolution}\label{specialres}
\tindex{Gresolution@$G$-resolution}

We close this chapter with the discussion of a special resolution
which computes bounded cohomology. This resolution will be one of the
ingredients needed to show that singular homology and measure
homology are isometrically isomorphic (see Section~\ref{isometricisom}).

\begin{defi}\label{specialdef}
  Let $X$ be an arcwise connected space with universal covering $\ucov
  X$, and let $k \in \N$.
  \begin{itemize}
    \item \dindex{group action!on $\map[norm]{\protect\ucov X^{k+1}}{\R}$}
          Then $\pi_1 (X)$ acts from the left on the vector 
          space~$\map[norm]{\ucov X^{k+1}}{\R}$ of continuous functions
          $\ucov X ^{k+1} \longrightarrow \R$ via
          \[ g \cdot f := \bigl( (x_0, \dots, x_k) \mapsto
                                 f(x_0 \cdot g,  \dots, x_k \cdot g)
                          \bigr) \]
          for all $f \in \map[norm]{\ucov X ^{k+1}}\R$ and all $g \in
          \pi_1(X)$.
    \item \nindex{F}{$\bbch k X$}{set of bounded functions in~$\map[norm]{\ucov
          X^{k+1}}\R$}% 
          \nindex{I}{$\bfch * X$}{set of $\pi_1(X)$-invariant functions
          in $\bbch * X$}% 
          The subset of bounded functions
          in~$\map[norm]{\ucov X^{k+1}}\R$ is denoted by~$\bbch * X$ 
          and we use the abbreviation
          $\bfch k X$ for the functions in~$\bbch k X$ which are invariant  
          under the above $\pi_1(X)$-action. 
     \qedhere
  \end{itemize}
\end{defi}

How can we turn $\bbch * X$ into a cochain complex? The vector space $\bchn[norm] k
{\ucov X}$ can be identified with the space of bounded
functions $\ssing[norm] k {\ucov X} \longrightarrow \R$ (under this identification,
the norm $\supn{\, \cdot\,}$ just becomes the supremum norm). 
 
Now $\bbch k X$ can be viewed as a subspace of~$\bchn[norm] k {\ucov X}$,
namely as the space of those bounded functions $\ssing[norm] k {\ucov X}
\longrightarrow \R$ which only depend on the vertices of the simplices
(and are continuous in the vertices). In Gromov's terminology those
functions would be called ``straight bounded continuous cochains''
\cite[Section~2.3]{gromov}.

It is clear that the coboundary operator on~$\bchn[norm] * {\ucov X}$
restricts to $\bbch * X$ and that the operations of $\pi_1(X)$ on
$\bbch * X$ and
$\bchn[norm] * {\ucov X}$ are compatible with the above inclusion
map. This makes $\bbch * X$ a subcomplex of~$\bchn[norm] * {\ucov
X}$. In other words, the homomorphism
\nindex{u}{$u^*$}{}
\begin{align*}
  u^k \colon \bbch k X & \longrightarrow \bchn[norm] k {\ucov X} \\
                     f & \longmapsto
                         \bigl( \sigma \mapsto
                                f(\sigma(e_0), \dots, \sigma(e_k))
                         \bigr)
\end{align*}
is a $\pi_1(X)$-equivariant cochain map.

\begin{satz}\label{specialthm}%
  \tindex{isometric isomorphism}%
  \tindex{Gresolution@$G$-resolution}
  Let $X$ be a connected countable \cw-complex and let $\ucov X$ be
  its universal covering space. 
  \begin{enumerate}
    \item Then the homomorphism
          \begin{align*}
            \varepsilon \colon \R & \longrightarrow 
                        \bbch 0 X = \map[norm]{\ucov X}\R \\
                                c & \longmapsto (x \mapsto c)
          \end{align*}
          turns the cochain complex 
          $\bbch * X$ into a strong relatively injective resolution
          by bounded $\pi_1(X)$-modules of the trivial
          $\pi_1(X)$-module $\R$.
    \item Hence the cohomology of $\bfch * X$ is canonically
          isomorphic to~$\bch * {\pi_1(X)}$. Moreover, under this
          isomorphism the supremum norm on~$\bbch * X$ induces the
          canonical seminorm on~$\bch *{\pi_1(X)}$. 
  \end{enumerate}
\end{satz}

\begin{qedkoro}\label{specialcor}%
  \tindex{isometric isomorphism}
  Let $X$ be a connected countable \cw-complex. Then $H^*\bigl(\bfch *
  X\bigr)$ and $\bch *X$ are isometrically isomorphic (by Theorem~\ref{bcandfg}).
\end{qedkoro}

A proof of the theorem is given in Monod's book 
\cite[Theorem~7.4.5]{monod}. However, Monod does not explicitly write
down a cochain map 
\[ \bfch * X \longrightarrow \bchn *X\] 
inducing this isometric isomorphism. 
Since we need a concrete description of this map in
Section~\ref{isometricisom}, we will now construct such a cochain map
with the help of the uniqueness part of Theorem~\ref{fundth}. 

As mentioned in Section~\ref{mtsec}, 
the bounded cohomology groups $\bch * {\pi_1(X)}$ can be computed via the
strong relatively injective resolution~$\bchn[norm]*{\ucov
X}$. Therefore we will compare the two complexes $\bfch *{X}$ and
$\bchn[norm]*{\ucov X}$: 
The ``inclusion'' map~$u^*$ from above fits into the ladder 
\begin{align*}
  \xymatrix@=2em{
      {0}  \ar[r]             
    & {\R} \ar[r]^-\varepsilon \ar[d]_{\id}
    & {\bbch 0X} \ar[r]        \ar[d]_{u^0}
    & {\bbch 1X} \ar[r]        \ar[d]_{u^1}
    & {\cdots} \\
   % --------------------------------
   {0} \ar[r] & {\R} \ar[r]_-\varepsilon
              & {\bchn[norm]0{\ucov X}} \ar[r] 
              & {\bchn[norm]1{\ucov X}} \ar[r]
              & {\cdots .}
  }
\end{align*}
Now Corollary~\ref{bcgcor} implies that (after taking the
$\pi_1(X)$-fixed points) $u^*$ induces the canonical isomorphism 
\[ H^*\bigl( \bfch * X\bigr) \longrightarrow 
   H^*\bigl(\bchn[norm]*{\ucov X}^{\pi_1(X)}\bigr) \]
(which is isometric by Theorem~\ref{specialthm}).
As mentioned in Section~\ref{mtsec}, the isometric isomorphism
\[ \widehat p^*  \colon \bch *X \longrightarrow 
   H^*\bigl(\bchn[norm]*{\ucov X}^{\pi_1(X)} \bigr)\]
is induced by the universal covering map~$p \colon \ucov X
\longrightarrow X$.
Thus the desired isometric isomorphism $H^*\bigl(\bfch * X\bigr)
\longrightarrow \bch * X$ is the homomorphism~$H^*(w^*)$ making the triangle 
\begin{align*}
  \xymatrix@R=2.5em@C=0.7em{
    {H^*\bigl( \bfch * X\bigr)} \ar[rr]^{H^*(w^*)} \ar[rd]_{H^*(u|^*)}
    & & {\bch * X} \ar[dl]^{\widehat p^*} \\
     & {H^*\bigl( \bchn[norm]*{\ucov X}^{\pi_1(X)} \bigr)}& 
  }
\end{align*}
commutative (where $u|^*$ denotes the restriction of~$u^*$ to the
$\pi_1(X)$-fixed points). 

The inverse of~$\widehat p ^*$ on the level of cochains can be
described as follows: 
for all $f \in \bchn[norm]k{\ucov X}^{\pi_1(X)}$ and $\sigma \in \ssing k X$ we
define
\[ \bigl(v^k(f)\bigr)(\sigma) := f(\ucov \sigma),\]
where $\ucov\sigma$ is some lift of~$\sigma$ with respect to~$p$. This
definition does not depend on the chosen lift since $f$ is
$\pi_1(X)$-invariant and any two lifts of~$\sigma$ only differ by
multiplication with an element in~$\pi_1(X)$ \cite[Lemma~V~8.1]{massey}. 
Since $f$ is bounded, $v^k(f)$ is bounded on~$\ssing k X$. Therefore linear
extension yields $v^k(f) \in \bchn k X$.

Covering theory implies that for all $\sigma \in \ssing k X$ and all
$j \in \{0, \dots, k\}$ each lift of the $j$-th face of~$\sigma$ 
is also the $j$-th face of some lift of~$\sigma$. Hence 
\[ v^* \colon \bchn[norm] * {\ucov X}^{\pi_1(X)} \longrightarrow 
   \bchn * X\]
is a cochain map. By construction, it is the inverse of the cochain
map $\bchn * X \longrightarrow \bchn[norm]*{\ucov X}^{\pi_1(X)}$
induced by the universal covering map~$p$.

Therefore, the composition
\[ w^* := v^* \circ u|^* \colon \bfch * X \longrightarrow \bchn *X\]
induces the canonical isometric isomorphism $H^*(\bfch * X ) \longrightarrow
\bch * X$. By definition, for all $f \in \bfch k X$ and all $\sigma
\in \ssing k X$ we obtain
\[ \bigl(w^k(f)\bigr)(\sigma) 
   = f \bigl( \ucov\sigma(e_0), \dots,\ucov\sigma(e_k)\bigr), \]
where $\ucov \sigma$ is any lift of~$\sigma$. This is the
description we will use in Section~\ref{isometricisom}.

diploma_main.tex
%&biglatex
% WWU Muenster 01/02/2004
% Main file
\documentclass[final,date,omega,math,twocoltoc]{diplomega} 

  \usepackage{amsmath}
  \usepackage{amsfonts}
  \usepackage{amssymb}

  \usepackage{url}
  \usepackage{nicefrac}
%  \usepackage{colortbl}

  \usepackage[final]{epsfig}
  \usepackage[final]{graphicx}
  \usepackage{tabularx}
  \usepackage[small]{caption}
  \usepackage[all,poly,dvips]{xy}
  \CompileMatrices
  \SelectTips{cm}{}

  \usepackage{pifont}

% -----------------------------------------
% Hyphenations ----------------------------

\hyphenation{asphe-ri-cal}
\hyphenation{In-vari-ants}
\hyphenation{in-tui-tively}
\hyphenation{ame-na-ble}
\hyphenation{Lip-schitz}

% index
\makeindex
% table of notation
\newindex{not}{ndx}{nnd}{Table of Notation}

% ------------------------------------------
% some repeatedly ocurring thms
% Gromov's conjecture on l2
\newcommand{\Gromovconjecture}{%
  \begin{conj}
    Let $M$ be an aspherical closed orientable manifold %of positive
    %dimension 
    whose simplicial volume vanishes. Then for all~$p \in \N$,
    \[ \ltb[big]{p}{\ucov{M};{\mathcal N}(\pi_1(M))} = 0.\]
 \end{conj}%
}
% l2 proportionality principle
\newcommand{\ltproptmp}{%
  Let $U$ be a simply connected Riemannian manifold. Then 
  for each $p \in\N$ there are 
  constants~$B_p \in \R_{> 0}$ and a constant~$T \in \R_{>0}$
  satisfying: for any discrete group~$G$ acting
  freely and cocompactly on~$U$ by isometries and all~$p \in \N$, 
  \begin{align*}
    \ltb[big]{p}{U; 
                  {\mathcal N}(G)} 
   = B_p \cdot \vol(G\setminus U).
  \end{align*}
  If $U$ with this $G$-action is of determinant class, then
  \[ \ltt[big]{U; {\mathcal N}(G)} = T \cdot \vol(G\setminus U).\]
}
\newcommand{\ltpropprincunn}{%
   \begin{satzunn}[Proportionality Principle of $L^2$-Invariants]
     \ltproptmp
   \end{satzunn}
}

\newcommand{\ltpropprinc}{%
   \begin{satz}[Proportionality of $L^2$-Invariants]
     \ltproptmp
   \end{satz}
}
% simplicial volume proportionality principle
\newcommand{\svproptmp}{%
  Let $M$ and $N$ be oriented closed connected Riemannian manifolds
  %(of positive dimension) 
  with isometrically isomorphic universal
  Riemannian coverings. Then 
  \[ \frac{\sv M}{\vol (M)} = \frac{\sv N}{\vol (N)}.\]}
\newcommand{\svpropprinc}{%
  \begin{satz}[Proportionality Principle of Simplicial Volume]
    \svproptmp
  \end{satz}
}
\newcommand{\svpropprincunn}{%
  \begin{satzunn}[Proportionality Principle of Simplicial Volume]
    \svproptmp
  \end{satzunn}
}
% hyperbolic proportionality principle
\newcommand{\hypproptmp}{%
  For $n \in\N$ define $v_n \in \R_{>0}$ as the maximal volume of an
  ideal $n$-simplex in $n$-dimensional hyperbolic space. 
  Then for each hyperbolic oriented closed connected manifold~$M$ of
  dimension~$n$, 
  \[ \frac{\sv M}{\vol(M)} = \frac1{v_n}. \]
}

\newcommand{\hyppropprincunn}{%
  \begin{satzunn}\hypproptmp
  \end{satzunn}
}
\newcommand{\hyppropprinc}{%
  \begin{satz}\hypproptmp
  \end{satz}
}

%-------------------------------------------

% ------------------------------------------
% BEGIN ------------------------------------
\begin{document}
\frontmatter


%----  TITEL -------------------------------
%-------------------------------------------

\begin{titlepage}
  \rule{\linewidth}{0pt} % door ks, tsts :)
  \vfill
  \begin{flushright}
    \Huge{The Proportionality Principle\\
          of Simplicial Volume}

    \vspace*{1em}

    \Large{Clara Strohm}
  \end{flushright}
  \vfill\vfill
\end{titlepage}
\thispagestyle{empty}
\begin{titlepage}
  \newlength{\wolf}
  \settowidth{\wolf}{Prof.\ Dr.\ Wolfgang L\"uck}
  \rule{\linewidth}{0pt} % door ks, tsts :)
  \begin{flushright}
    Westf\"alische Wilhelms-Universit\"at M\"unster \\
    Fachbereich Mathematik und Informatik
  \end{flushright}
  \vfill
  \begin{flushright}
    \Large{Diplomarbeit}

    \vspace*{1em}

    {\Huge The Proportionality Principle\\
           of  Simplicial Volume}

    \vspace*{1em}

    {\Large Clara Strohm}

    \vspace*{.5em}
     
    \normalsize{September 2004}
   \end{flushright}
   \vfill\vfill
   \begin{flushright}
     \begin{tabular}{rr}
       Betreuer :      & Prof.\ Dr.\ Wolfgang L\"uck \\
       Zweitgutachter: & \makebox[\wolf][s]{PD\ Dr.\ Michael Joachim}
     \end{tabular}
   \end{flushright}
\end{titlepage}
\thispagestyle{empty}
%  ---------------------------------
%  Selbstaendigkeitserklaerung
%  ---------------------------------
\rule{\linewidth}{0pt} % door ks, tsts :)

\vfill

Hiermit versichere ich, da\ss\ ich diese Arbeit selbst\"andig verfa\ss t
und nur die angegebenen Quellen und Hilfsmittel benutzt habe. 

\vspace*{1em}

M\"unster, im September 2004

\chapter*{Introduction}

Manifolds, the basic objects of geometry, form the playground for both
topological and smooth structures. Many challenges in modern
mathematics are concerned with the nature of this interaction between
algebraic topology, differential topology and differential
geometry. For example all incarnations of the index theorem, 
the Gau\ss-Bonnet formula, 
and the Mostow
rigidity theorem fall into this category. 

\begin{satzunn}[Mostow Rigidity Theorem]\tindex{Mostow rigidity theorem}%
  Each homotopy equivalence between 
  oriented closed connected hyperbolic manifolds of dimension at least~3
  is homotopic to an isometry. 
\end{satzunn}

This theorem was first proved by Mostow \cite{mostow}.  
But there is also a proof by Gromov \cite{munkholm}, for which he
designed the simplicial volume (therefore also known as ``Gromov
norm'').

The simplicial volume is a homotopy invariant of oriented closed
connected manifolds defined in terms of the singular chain complex
with real coefficients. This invariant consists of a 
nonnegative real number measuring the efficiency of representing
the fundamental class using singular simplices: 
If $M$ is an oriented closed connected manifold, then
the \emph{simplicial volume} of~$M$, denoted by~$\sv M$, is the infimum of
all $\ell^1$-norms of singular cycles with real coefficients representing the
fundamental class.
Since the fundamental class can be viewed as a
generalised triangulation of the manifold, the simplicial volume can
also be interpreted as a measure for the complexity of a manifold. 

But the simplicial volume also has another important interpretation:
it is a homotopy invariant approximation of the Riemannian volume. For
example, 
Gromov's proof of the Mostow rigidity theorem is based on the
following proportionality principle for hyperbolic manifolds 
\cite[Section~2.2]{gromov}, \cite[Theorem~6.2]{thurston}:


\hyppropprincunn

In particular, the simplicial volume does not always vanish.  However,
in general the simplicial volume is rather hard to compute. In the
most cases where the simplicial volume is known, it is zero.  This,
for example, is the case if the fundamental group of the manifold is
Abelian or, more generally, amenable.

One of the most complete (but hardly digestible) references about the
simplicial volume is Gromov's pioneering paper ``Volume and
Bounded Cohomology'' \cite{gromov}. More recent introductions are
given in the textbooks on hyperbolic geometry by Ratcliffe and
Benedetti, Petronio \cite[\S11.5]{ratcliffe}, \cite[Section~C.3]{bp}.

Applications of the simplicial volume also include knot theory
\cite{murakami}, the existence of certain $S^1$-operations
\cite{yanos1}, and the investigation of other volumes, such as the
Riemannian or the minimal volume \cite{gromov}.  
Furthermore, Gromov conjectured the following relation between the
simplicial volume and $L^2$-invariants \cite[Section 8A]{gromovasym}
-- thereby connecting two distant fields of algebraic topology:

\Gromovconjecture

A rather complete account of $L^2$-invariants is given in the book
\emph{$L^2$-Invariants: Theory and Applications to Geometry and
$K$-theory} \cite{lueck}. The above conjecture is justified by the 
observation that $L^2$-invariants and the simplicial volume show a similar
behaviour. We will discuss this conjecture briefly in
Section~\ref{l2sec}. Among the most striking similarities is the
proportionality principle:

\ltpropprincunn

In particular, if $M$ and $N$ are oriented closed connected Riemannian
manifolds with isometrically isomorphic universal Riemannian
coverings, then for all $p \in \N$,
\[ \frac{\ltb[big]{p}{\ucov M; {\mathcal N}(\pi_1(M))}}{\vol(M)}
 = \frac{\ltb[big]{p}{\ucov N; {\mathcal N}(\pi_1(N))}}{\vol(N)}
\]
and
\[ \frac{\ltt[big]{\ucov M; {\mathcal N}(\pi_1(M))}}{\vol(M)}
 = \frac{\ltt[big]{\ucov N; {\mathcal N}(\pi_1(N))}}{\vol(N)}.
\]

\svpropprincunn

These proportionality principles are also examples for an interesting
link between topological and differential structures. A similar
proportionality principle is known for Chern numbers
\cite[Satz~3]{hirzebruch}. 

A proof in the $L^2$-case is given in L\"uck's book
\cite[Theorem~3.183]{lueck}.  For the simplicial volume, sketches of
(different) proofs were given by Gromov \cite[Section~2.3]{gromov} and
Thurston \cite[page~6.9]{thurston}.  However, there is no complete
proof of the proportionality principle of simplicial volume in the
literature. It is the aim of this diploma thesis to provide such a
proof, based on Thurston's approach.

This text is organised as follows:

In the first chapter the simplicial volume is introduced and some of
its properties are collected.

Chapter~2 is devoted to the study of bounded cohomology, a quite
peculiar variant of singular cohomology. Its application as a basic
tool for the analysis of the simplicial volume was discovered by
Gromov \cite{gromov}.  However, we will mostly refer to the more
elegant approach due to Ivanov, based on classical homological algebra
\cite{ivanov}, \cite{monod}.

Thurston's proof of the proportionality principle relies on the
computation of the simplicial volume by means of a new homology theory
called measure homology \cite[pages~6.6--6.9]{thurston}.  Instead of
linear combinations of singular simplices, measures on the set of
(smooth) singular simplices are considered as chains, thereby
permitting averaging constructions such as Thurston's smearing.  But
Thurston's exposition lacks a proof of the fact that measure homology
and singular homology give rise to the same simplicial volume.  We
will define measure homology in Chapter~3 and list its basic algebraic
properties as they can be found in the papers of Zastrow and Hansen 
\cite{zastrow}, \cite{hansen}, in
a slightly different setting. 

Chapter~4 constitutes the central contribution of this diploma
thesis. It will be shown that measure homology indeed can be used to
compute the simplicial volume, i.e., that singular homology with real
coefficients is \emph{isometrically} isomorphic to measure
homology. The proof is based on techniques from bounded cohomology, as
explained in Chapter~2. It will also be discussed why Bowen's argument
\cite{bowen} is not correct (Subsection~\ref{bowen}).

The last chapter finally contains the proof of the proportionality
principle -- using Thurston's smearing technique. This forces us to
study isometry groups of Riemannian manifolds.  En passant, it will be
proved that the compact open topology and the \cone-topology (the
smooth counterpart of the compact open topology) on the isometry
groups coincide. Unfortunately, I was not able to show this without
making use of the theory of standard Borel spaces.  The chapter closes
with a proof of the fact that the simplicial volume of a flat
orientable closed connected manifold vanishes. The proof is based on
the proportionality principle, instead of referring to Gromov's
sophisticated estimate of the simplicial volume via the minimal volume
\cite[page~220]{gromov}.

I tried to keep the prerequisites for the understanding of this diploma
thesis as small as possible. So only a basic familiarity with singular
(co)homology, covering theory, measure theory, and Riemannian geometry
is required. A variety of textbooks is available providing such a
background: e.g., \cite{lee}, \cite{leesmooth}, \cite{massey}, \cite{bredon},
\cite{elstrodt}. But the detailed exposition of the subject comes at a
price -- the length of the text, for which I apologise.

The symbols $\Z$, $\N$, $\Q$, $\R$, $\C$ stand for the set of integers,
the set of nonnegative integers, the set of rational numbers, the set
of real numbers, and the set of complex numbers respectively. The one
point space is written as~$\opsp$. For simplicity, we always assume
that manifolds have dimension at least~1. The (co)homology class
represented by an element~$x$ is denoted by~$[x]$. 
All sums ``$\sum$'' are implicitly finite, unless stated otherwise. 
All notation is
collected in the table of notation (between bibliography and index). 

% basic notation:
\snindex{$a}{$\consum$}{connected sum}%
\snindex{$b}{$\opsp$}{one point space}%
\snindex{$c}{$\lor$}{wedge}%
\snindex{ker}{$\ker$}{kernel}%
\snindex{im}{$\im$}{image}%
\snindex{$s}{$[x]$}{(co)homology class represented by~$x$}%
\snindex{vol}{$\vol(M)$}{volume of~$M$}%
\snindex{pi1}{$\pi_1$}{fundamental group}%
\snindex{Tf}{$Tf$}{differential of~$f$}%
\snindex{volM}{$\vol_M$}{volume form of~$M$}%
\snindex{C}{$\C$}{complex numbers}%
\snindex{N}{$\N$}{nonnegative integers}%
\snindex{Q}{$\Q$}{rational numbers}%
\snindex{R}{$\R$}{real numbers}%
\snindex{Z}{$\Z$}{integers}%
\snindex{colim}{$\dirlim$}{colimit/direct limit}%
\vspace*{-\baselineskip}
\section*{Acknowledgements}
\enlargethispage{\baselineskip}

First of all I would like to thank Professor Wolfgang L\"uck for
giving me the opportunity to write this diploma thesis, for the 
encouragement, and for the freedom I enjoyed during this project.

Roman Sauer and Marco Schmidt infected me with their fascination for
the simplicial volume and bounded cohomology. I am very grateful for
all their support and guidance.  Marco Varisco was always open for
discussions on mathematical principles and notation, and for making
XYZ-theory accessible for pedestrians like me. Mille grazie!

Professor Volker Puppe and his concise topology lectures and
seminars led me through my two years at the University of 
Konstanz. I would like to thank him  for all his support. 

Furthermore, I would like to thank Stefanie Helker for studying English and 
Benedikt Plitt, 
Julia Weber (Wohlf\"uhlb\"uro!), and 
Michael Weiermann 
for hunting bugs and for their valuable suggestions.

My parents always supported me, even though I did not manage to 
become acquainted with philosophy -- but at least I am no 
\foreignlanguage{polutonikogreek}{>agewm'etrhtoc} 
anymore. Dankesch\"on!

All my gratitude is devoted to Andres L\"oh, not only for his 
\TeX nical ekspertise, but also for his patience and all the things 
language is not powerful enough to express -- except for
Pl\"uschtiere: Wuiisch? Zonder pf!
% end intro
% -------------------------------------------------------

% -------------------------------------------------------
% tableofcontents
% auf Doppelseite!
\begingroup
\makeatletter
\let\@mkboth=\@gobbletwo
\makeatother
\let\cleardoublepage=\clearpage
\tableofcontents
\endgroup

% ------------------------------
%
\mainmatter

%--------------------------------------------------------------------
\chapter{Simplicial Volume}


\begin{chapterabstract} 
 The simplicial volume of an oriented closed connected manifold is a
 homotopy invariant -- defined in terms of the singular chain complex -- 
 which measures the efficiency of representing the fundamental class by
 singular chains (with real coefficients). Its first appearance is in
 Gromov's famous proof of the Mostow rigidity theorem
 \cite{munkholm}. 

 Despite of being a topological invariant the simplicial volume contains
 interesting information about the possible differential structures on
 a manifold -- in form of the Riemannian volume. Examples for such
 relations are Gromov's estimate of the minimal volume and the
 proportionality principle, which is the topic of this thesis.  
 
 Moreover, there seems to be a deep connection to  
 $L^2$-invariants \cite[Section~8A]{gromovasym}:
 
 \Gromovconjecture
  
 In this chapter the simplicial volume is introduced
 (Section~\ref{svdefsec}) and some of its properties are collected
 (Section~\ref{svpropsec}).  We then give a short survey about
 indications supporting the above conjecture in Section~\ref{l2sec}.
 In the last section we briefly discuss some generalisations of the
 simplicial volume.
\end{chapterabstract}

\InputIfFileExists{simvol}{}

% -------------------------------------------------------------------

\chapter{Bounded Cohomology}

\begin{chapterabstract}\label{chboco}
 Bounded cohomology is the functional analytic twin of singular
 cohomology -- it is constructed similarly to singular cohomology from
 the singular chain complex, using the topological dual space instead of the
 algebraic dual space \cite{gromov}. The corresponding duality on the
 level of (co)homology is explored in Section~\ref{dualitysection}.
 Using this duality, it is possible to calculate the simplicial volume
 via bounded cohomology.
 
 It is astonishing that the rather small difference between the
 definitions of singular and bounded cohomology lead to
 completely different characters of both theories. 

 Apart from the above geometric definition, there is also a notion of
 bounded cohomology of groups due to Ivanov based on techniques from
 homological algebra \cite{ivanov}, which we will sketch in
 Section~\ref{bcgroupssection}. This algebraic approach is the source
 of the strength of bounded cohomology.

 In Section~\ref{mtsec}, a key feature of bounded cohomology is
 discussed: the bounded cohomology of a topological space coincides
 with the bounded cohomology of its fundamental group.  Unfortunately,
 unlike singular cohomology bounded cohomology fails to satisfy the
 excision axiom. Nevertheless, in many cases bounded cohomology can be
 calculated directly from special resolutions. For example, it can be
 shown that bounded cohomology ignores amenable groups, implying that
 the bounded cohomology of spaces with amenable fundamental group
 vanishes.
 
 In the last section, we will explain a special resolution for
 calculating the bounded cohomology of a group, which plays a crucial
 r\^ole in Section~\ref{isometricisom}. 
\end{chapterabstract}
  

\InputIfFileExists{boco}{}

\addtocontents{toc}{\protect\end{multicols}}
\addtocontents{toc}{\protect\newpage}
\addtocontents{toc}{\protect\rule{\linewidth}{0pt}}
\addtocontents{toc}{\protect\vspace*{12em}}
\addtocontents{toc}{\protect\begin{multicols}{2}}
\chapter{Measure Homology}\label{chmh}

\begin{chapterabstract}
  Measure homology is a generalisation of singular homology in the
  following way: singular chains with real coefficients are viewed
  as signed measures on the space of singular simplices whose mass is
  concentrated in finitely many points. In the measure homology chain 
  complex also more complicated measures are allowed.

  Measure homology was introduced in Thurston's lecture notes
  in~1979 \cite[page~6.6]{thurston}. He already claimed that
  measure homology and singular homology should coincide. 
  More extensive accounts are the papers of Zastrow \cite{zastrow} and
  Hansen \cite{hansen} and the book of Ratcliffe \cite{ratcliffe}.
  
  The motivation for the introduction of measure homology originates
  from the fact that measure homology can be used to calculate the
  simplicial volume, hence giving more room for constructions such as
  smearing (cf.~Chapter~\ref{chpropprinc}). This smearing construction
  will be the key to the proof of the proportionality principle.

  In Section~\ref{prelude}, the technical background needed for the
  definition of measure homology is presented -- in particular, a
  feasible topology on the set of smooth singular simplices will be
  introduced. In Section~\ref{mhdefsec} measure homology is
  defined. The basic properties of measure homology are listed in
  Section~\ref{mhprop}, and there is also given a proof for the
  compatibility of measure homology with colimits.  The next chapter
  is devoted to the proof that measure homology and singular homology
  are isometrically isomorphic.
\end{chapterabstract}

\InputIfFileExists{meashom}{}


% -------------------------------------------------------------------

\chapter{The Proportionality Principle}\label{chpropprinc}

\begin{chapterabstract}
  The proportionality principle of simplicial volume reveals a
  fascinating connection between the simplicial volume and the
  Riemannian volume: simplicial volume and Riemannian volume are
  proportional if we restrict our attention to manifolds which share
  the same universal Riemannian covering space. 
  Similar proportionality principles also occur in the setting of
  $L^2$-invariants (Theorem~\ref{l2pp}) and in the setting of
  characteristic numbers of complex manifolds
  \cite[Satz~3]{hirzebruch}. 

  Both Thurston and Gromov sketched (dual) proofs for the
  proportionality principle of simplicial volume \cite[page~6.9]{thurston},
  \cite[Section~2.3]{gromov}. Thurston's idea is to use measure
  homology to compute the simplicial volume and to take advantage of
  the larger chain complex for the so-called smearing construction.
  In this chapter, a detailed version of Thurston's proof will be
  given, based on the results of the previous chapters.

  Gromov's proof uses arguments from bounded cohomology and also
  depends on some averaging operation via Haar measures. Unfortunately his
  exposition is not very explicit about certain measurability issues. 
 
  In Section~\ref{stpp}, the proportionality principle is discussed and
  the strategy of proof is presented. Sections \ref{isometrygroups}
  and \ref{haarmeasure} provide the necessary tools for Thurston's
  smearing technique (i.e., the study of isometry groups and
  appropriate measures on them).  In Section~\ref{smearing}, the
  smearing of smooth singular chains is developed. The actual proof
  of the proportionality principle is given in
  Section~\ref{pproof}. The chapter ends with some applications in
  Section~\ref{ppapp}.
\end{chapterabstract}

\InputIfFileExists{propprinc}{}

% -----------------------------------------------------------------
% literatuur ------------------------------------------------------

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\end{thebibliography}

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\cleardoublepage
\printindex

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meashom.tex
% part of omegas diploma thesis
% main: diploma_main.tex
\section{Prelude}\label{prelude}

In this section the basic objects involved in the definition of
measure homology are examined.  The first subsection introduces signed
measures. In the later subsections, topologies for mapping spaces are
provided and some technicalities concerning smooth maps on standard
simplices are discussed.

% -------------------------------------------------------------------
\subsection{Signed measures}

Signed measures are a slight generalisation of positive measures. As the 
name suggests, signed measures can also take negative values. The 
generalisation is quite straightforward; however, some additional care 
has to be taken when defining null sets or integrability with respect 
to a signed measure. The main advantage of signed measures is that 
the set of all finite signed measures on a given measurable space is a 
vector space over the real numbers. 

\begin{defi}
  Let $(X, \siga)$ be a measurable space. 
  \begin{itemize}
    \item  \dindex{signed measure}%
           A map $\mu \colon \siga \longrightarrow 
           \R \cup \{ \infty, -\infty\}$ is called a \defin{signed measure} if
           $\mu(\emptyset) = 0$, not both $\infty$ and $-\infty$ are contained 
           in the image of $\mu$, and $\mu$ is $\sigma$-additive.
    \item  \dindex{null set}%
           A \defin{null set} of a signed measure $\mu$ is a measurable 
           set~$A$ with $\mu(B) = 0$ for all~$B \in \siga$ 
           with~$B \subset A$.
    \item  \dindex{mupositive@$\mu$-positive}%
           \dindex{mungeative@$\mu$-negative}%
           A set $A \in \siga$ is called $\mu$\defin{-positive} if 
           $\mu(B) \geq 0$ for all $B \in \siga$ with~$B \subset A$. 
           Analogously, $\mu$\defin{-negative} sets are defined. 
    \item  \dindex{determination set}%
           A \defin{determination set of}~$\mu$ is a subset~$D$ 
           of~$X$ such that each measurable set contained in the complement 
           of $D$ is a $\mu$-null set. 
           \qedhere
  \end{itemize}
\end{defi}

Of course, each ordinary positive measure is also a signed measure. If 
$\mu$ and~$\nu$ are two positive measures, at least one of them being 
finite, then $\mu - \nu$ and $\nu - \mu$ are two signed measures. In fact, 
these are the \emph{only} examples since each signed measure can 
be written as such a difference. 

\begin{satz}[Hahn Decomposition]
  \dindex{Hahn decomposition}%
  \dindex{decomposition!Hahn}%
  Let $\mu$ be a signed measure on the measurable space $(X,\siga)$. Then there 
  is a decomposition $X = P \cup N$ where $P$ is a $\mu$-positive set 
  and $N$ is a $\mu$-negative set.
  
  This decomposition is essentially unique: if $X = P' \cup N'$ is
  another decomposition in a $\mu$-positive and a $\mu$-negative set,
  then the symmetric difference of $P$ and $P'$ (which equals the
  symmetric difference of $N$ and $N'$) is a $\mu$-null set.
\end{satz}

A proof is, for example, given in Elstrodt's textbook on measure
theory and integration \cite[Satz 1.8 in Kapitel VII]{elstrodt}. 

\begin{defi}
  Let $\mu$ be a signed measure on $(X,\siga)$ and $X = P \cup N$ its Hahn 
  decomposition. 
  \begin{itemize}
    \item \dindex{positive variation}%
          \dindex{variation!positive}%
          \dindex{negative variation}%
          \dindex{variation!negative}%
          The (positive) measure
          \begin{align*} 
            \mu^+ \colon \siga & \longrightarrow \R_{\geq 0} \cup\{ \infty\} \\
                         A     & \longmapsto     \mu(A \cap P)
          \end{align*}
          is called \defin{positive variation of}~$\mu$.
          Analogously, the restriction $\mu^-$ of $-\mu$ to the set $N$ is 
          called \defin{negative variation of}~$\mu$.
          \dindex{Jordan decomposition}%
          \dindex{decomposition!Jordan}% 
          The decomposition $\mu = \mu^+ - \mu^-$ is the \defin{Jordan 
          decomposition of}~$\mu$. 
    \item \dindex{variation}%
          The \defin{variation of}~$\mu$ is defined as the sum
          \[ |\mu| := \mu^+ + \mu^-\]
          \dindex{total variation}%
          \dindex{variation!total}%
          \nindex{$la}{$\tov{\cdot}$}{total variation of a signed measure}
          and the \defin{total variation of}~$\mu$ is given by  
          \[ \tov{\mu} := |\mu|(X) \in \R_{\geq 0}\cup \{\infty\}.\qedhere\]
  \end{itemize} 
\end{defi}

Since the Hahn decomposition is unique up to null sets, the positive and the 
negative variation of a signed measure are indeed well-defined. Moreover, the 
sum $\mu^+ + \mu^-$ makes sense because at least one of these two 
measures must be finite.

Sometimes it is useful to have another description of the total 
variation:

\begin{lem} \label{tov}
  \tindex{total variation}
  \tindex{variation!total}
  Let $(X,\siga)$ be a measurable space and $\mu$ a signed measure on~$(X,\siga)$. 
  Then 
  \[ \tov{\mu} = \sup_{A \in \siga} \mu(A) 
               - \inf_{B \in \siga} \mu(B). \]
\end{lem}
\begin{bew}
  Let $X = P \cup N$ be the Hahn decomposition of~$\mu$. By definition,
  \[ \tov\mu = |\mu| (X)
             = \mu(P) - \mu(N). \]
  This clearly shows ``$\leq$''. On the other hand, for all $A$,~$B \in \siga$
  \begin{align*}
    \mu(A) - \mu(B) & =    \mu^+(A) - \mu^-(A)
                         - \mu^+(B) + \mu^-(B) \\
                    & \leq \mu^+(A)
                         + \mu^-(B) \\
                    & \leq \mu^+(X)
                         + \mu^-(X) \\ 
                    & =    |\mu|(X) \\
                    & =    \tov\mu
  \end{align*}
  since $\mu^+$ and $\mu^-$ are positive measures.
\end{bew}


\begin{defi}
  \dindex{Borel sigmaalgebra@Borel $\sigma$-algebra}%
  Let $X$ be a topological space.
  The \defin{Borel $\sigma$-algebra} is the $\sigma$-algebra on~$X$ 
  generated by all open sets of~$X$. 
\end{defi}

\begin{defi}
  Let $(X,\siga)$ be a measurable space and $f \colon (X,\siga) \longrightarrow \R$ 
  a measurable function (with respect to the Borel~$\sigma$-algebra on~$\R$).
  \dindex{integrable}%
  If $\mu$ is a signed measure on~$(X,\siga)$, we call $f$ 
  \defin{integrable with respect to}~$\mu$ if $f$ is integrable with 
  respect to the positive measures $\mu^+$ and~$\mu^-$ in the usual  
  sense. 
  \dindex{integral}%
  The \defin{integral of}~$f$ \defin{with respect to}~$\mu$ is  then 
  defined as
  \[ \int f \;d\mu := \int f \;d\mu^+ - \int f \;d\mu^-. \qedhere\]
\end{defi}

\begin{lem}\label{tovnorm}
  \tindex{total variation}%
  \tindex{variation!total}%
  Let $X$ be a topological space. The set of all signed measures on the 
  Borel~$\sigma$-algebra of~$X$ possessing a compact determination set and 
  finite total variation is a real vector space. The total variation turns 
  this vector space into a normed vector space.
\end{lem}
\begin{bew}
  Let $\mu$ and $\nu$ be two such measures and let $\alpha \in \R$. 
  Obviously, $\alpha \cdot \mu$ again is a signed measure. Each determination 
  set of~$\mu$ is a determination set of~$\alpha \cdot \mu$ and
  \[ \tov{\alpha \cdot \mu} = |\alpha| \cdot \tov \mu.\]

  Since both $\mu$ and $\nu$ have finite total variation, their value 
  on each Borel set is finite. 
  Hence the sum $\mu + \nu$ is well-defined and again a signed measure. 
  Furthermore, 
  if $D_\mu$ and~$D_\nu$ are (compact) determination sets of $\mu$ 
  and~$\nu$ respectively then $D_\mu \cup D_\nu$ is a (compact) 
  determination set of $\mu + \nu$.

  Using Lemma~\ref{tov} we get (where $\siga$ is the Borel~$\sigma$-algebra
  on~$X$) 
  \begin{align*}
    \tov{\mu + \nu} & =    \sup_{A \in \siga}\bigl( \mu(A) + \nu(A)\bigr)
                         - \inf_{B \in \siga}\bigl( \mu(B) + \nu(B)\bigr) \\
                    & \leq \sup_{A \in \siga} \mu(A)
                         + \sup_{A \in \siga} \nu(A)
                         - \inf_{B \in \siga} \mu(B)
                         - \inf_{B \in \siga} \nu(B) \\
                    & =    \tov \mu + \tov \nu.
  \end{align*}
  In particular, $\tov{\mu + \nu}$ is finite.
\end{bew}

\begin{defi}
  \dindex{Borel function}%
  Let $f \colon X \longrightarrow Y$ be a \defin{Borel function}, i.e., 
  measurable with respect to the Borel $\sigma$-algebra. Moreover, let $\mu$ be a signed 
  measure on the Borel~$\sigma$-algebra of~$X$. Then $\mu$ induces  
  a signed measure $\mu^f$ on~$Y$ given by
  \dindex{pushforward of a signed measure}%
  \dindex{induced measure}%
  \nindex{muf}{$\mu^f$}{pushforward of~$\mu$ with respect to~$f$}
  \[ \mu^f (A) := \mu\bigl( f^{-1} (A)\bigr) \]
  for all Borel subsets~$A \subset Y$.
\end{defi}

\begin{lem}\label{mhabschluss}
  Let $f \colon X \longrightarrow Y$ be a continuous function and 
  let $\mu$ be a 
  signed measure on the Borel~$\sigma$-algebra of $X$ possessing a compact 
  determination set and finite total variation. Then the induced 
  measure~$\mu^f$ also has a compact determination set and finite total 
  variation.
\end{lem}
\begin{bew}
  It is easy to see that continuous functions are Borel functions since the 
  Borel~$\sigma$-algebra is generated by the open sets. 
  Hence $\mu^f$ is well-defined.

  If $D$ is a compact determination set of~$\mu$, then the image $f(D)$ of 
  course is a determination set of~$\mu^f$. Since $f$ is continuous, $f(D)$ 
  is compact.
   
  Writing $\siga_X$ and $\siga_Y$ for the Borel~$\sigma$-algebras 
  on~$X$ and~$Y$ respectively we compute
  the total variation of~$\mu^f$ via Lemma~\ref{tov}
  as follows
  \begin{align*}
    \tov[big]{\mu^f} 
                & =    \sup_{A \in \siga_Y} \mu^f(A)
                     - \inf_{B \in \siga_Y} \mu^f(B) \\
                & =    \sup_{A \in \siga_Y} \mu\bigl(f^{-1}(A)\bigr)
                     - \inf_{B \in \siga_Y} \mu\bigl(f^{-1}(B)\bigr) \\
                & \leq \sup_{A \in \siga_X} \mu(A)
                     - \inf_{B \in \siga_X} \mu(B) \\
                & =    \tov{\mu}.
  \end{align*}
  Hence $\mu^f$ also has finite total variation.
\end{bew}

% -------------------------------------------------------------------  
\subsection{Mapping spaces}

Since we want to define measure chains as signed measures on some set of singular 
simplices, we have to introduce an appropriate topology on 
the corresponding mapping space. 
Unfortunately, the compact open topology is too coarse for our purposes. 
So we need another topology -- the so-called \cone-topology defined on
the space of smooth maps of smooth manifolds --, which takes into
account the smooth structure. 

\begin{defi}
  \nindex{map}{$\map X Y$}{set of continuous maps from~$X$ to~$Y$}%
  Let $X$ and $Y$ be two topological spaces and $\map{X}{Y}$ the set of 
  continuous maps~$X \longrightarrow Y$. 
  \begin{itemize}
    \item If $K \subset X$ is compact and $U \subset Y$ is open, we write
          \nindex{UK}{$U^K$}{}%
          \[ U^K := \{    f \in \map{X}{Y}
                    \,|\, f(K) \subset U
                    \}.\]
    \item \dindex{compact open topology}%
          \dindex{topology!compact open}%
          \dindex{topology!on map@on $\map X Y$}%
          The \defin{compact open topology} on the mapping space
          $\map{X}{Y}$ is the topology with the subbase
          \[ \{    U^K 
             \,|\, \text{$K \subset X$ compact, %
                         $U \subset Y$ open}    %
             \}.\qedhere\]
  \end{itemize}
\end{defi}

\begin{qedbem}\label{ev}
  In particular, for all $x \in X$ the evaluation map
  \begin{align*}
    \map X Y & \longrightarrow Y \\
	   f & \longmapsto     f(x)
  \end{align*}
  is continuous.
\end{qedbem}

\begin{defi}
  Let $M$ and $N$ be two smooth manifolds without boundary. 
  As sets $\maps{M}{N}$ and 
  \nindex{mapinf}{$\maps M N$}{set of all smooth maps from~$M$ to~$N$ with the
  \cone-topology}%
  \nindex{mapco}{$\mapsco M N$}{set of all smooth maps from~$M$ to~$N$ with the
  compact open topology}%
  $\mapsco{M}{N}$ are just the sets of smooth maps~$M \longrightarrow N$.
  \begin{itemize}
    \item \dindex{topology!on maps1@on $\mapsco M N$}%
          The topology on $\mapsco{M}{N} \subset \map{M}{N}$ is the
          subspace topology given by the compact open topology 
          on~$\map{M}{N}$.
    \item \dindex{topology!on maps2@on $\maps M N$}%
          The topology on $\maps{M}{N}$, called 
          \cone\defin{-topology}\dindex{c1topology@\cone-topology}%
          \dindex{topology!c1topology@\cone-topology}, 
          is the topology consisting of all inverse images of the 
          form~$T^{-1}(U)$ where $U$ is an open 
          subset of~$\map{TM}{TN}$ 
          and $T$ is the differential:
          \begin{align*}
            T \colon \maps{M}{N} & \longrightarrow \map{TM}{TN} \\
                               f & \longmapsto     Tf.
          \end{align*}
          I.e., the injective map $T \colon \maps{M}{N} \longrightarrow 
          \map{TM}{TN}$ is turned into a homeomorphism onto its 
          image~$T\bigl(\maps{M}{N}\bigr).$ \qedhere
  \end{itemize}
\end{defi}

More information about the \cone-topology and similar topologies can
be found in Hirsch's book \cite[Chapter~2]{hirschbook}. The above description is
taken from a paper of Hirsch \cite[page~244]{hirschpaper}. 

\begin{bem}\label{uniform}%
\tindex{compact open topology}%
\tindex{topology!compact open}%
\tindex{conetopology@\cone-topology}%
\tindex{topology!c1topology@\cone-topology}%
The compact open topology is the same as the topology of uniform
convergence on compact subsets \cite[Theorem~XII~7.2]{dugundji}.
Similarly, the \cone-topology 
is the topology of uniform convergence on compact 
subsets not only of the function itself but also of its differential.
\end{bem}

\begin{lem}\label{finer}
  Let $M$ and $N$ be two smooth manifolds without boundary.
  Then the \cone-topology on~$\maps{M}{N}$ is finer than the compact 
  open topology.
\end{lem}
\begin{bew}
  Denoting the zero section $M \hookrightarrow TM$ by~$i$ and the bundle 
  projection~$TN \longrightarrow N$ by $\pi$, we get for
  all compact sets $K \subset M$ and all open sets $U \subset N$ that
  \begin{align*}
   & T\bigl( U^K \cap \maps{M}{N} \bigr)\\ 
   & \qquad  = \bigl\{    Tf \colon TM \longrightarrow TN   
             \,\bigm|\, f \in U^K \cap \maps{M}{N}
               \bigr\} \\
   & \qquad  = \bigl\{    Tf \colon TM \longrightarrow TN   
             \,\bigm|\, f \in \maps{M}{N}, 
                          Tf(i(K)) \subset \pi^{-1}(U)
               \bigr\} \\   
   & \qquad  = \bigl\{    g \in \map{TM}{TN}
             \,\bigm|\,  g(i(K)) \subset \pi^{-1}(U)
               \bigr\}
             \cap
               T\bigl(\maps{M}{N}\bigr).
  \end{align*}
  Since $i$ and $\pi$ are continuous, $i(K) \subset TM$ is compact 
  and $\pi^{-1}(U) \subset TN$ is open. 
  Thus  
  the set $U^K \cap \maps{M}{N}$ is open in $\maps{M}{N}$.   
\end{bew}

\begin{lem}\label{comp}
  \begin{enumerate}
    \item Let $X$, $Y$, and $Z$ be topological spaces and $f \in \map{X}{Y}$, 
          $g \in \map Y Z$. Then the maps 
          \begin{align*}
            \map Y Z & \longrightarrow \map X Z\\
                   h & \longmapsto     h \circ f
          \end{align*}
          and
          \begin{align*}
            \map X Y & \longrightarrow  \map X Z \\
                   h & \longmapsto      g \circ h
          \end{align*}
          are continuous.
    \item Let $L$, $M$, $N$ be smooth manifolds without boundary 
          and $f \in \maps{L}{M}$,
          $g \in \maps{M}{N}$. Then the maps
          \begin{align*}
            \maps M N & \longrightarrow \maps L N \\
                    h & \longmapsto     h \circ f
          \end{align*}
          and
          \begin{align*}
            \maps L M & \longrightarrow \maps L N \\
                    h & \longmapsto     g \circ h
          \end{align*}
          are continuous.
  \end{enumerate}
\end{lem}
\begin{bew}
   1. This can easily be seen using the definition of the compact open 
      topology.

   2. Applying the first part to $Tf \in \map{TL}{TM}$ and $Tg \in 
      \map{TM}{TN}$ proves the second part. 
\end{bew}

% -------------------------------------------------------------------
\subsection{``Smooth'' maps on standard simplices}

Since the standard simplex $\ssim k$ is quite angular, some care has to 
be taken when defining the set of all smooth 
maps~$\ssim k \longrightarrow M$, where $M$ is some 
smooth manifold without boundary. 
The elements of $\maps[norm]{\ssim k}{M}$ should be thought of as being smooth 
maps on some slightly larger set without ``corners.'' More precisely:


\begin{defi} 
  Let $M$ be a smooth manifold without boundary and $k \in \N$. 
  \begin{itemize}
    \item The $k$-dimensional hyperplane in $\R^{k+1}$ spanned by 
          the vertices of the standard $k$-simplex~$\ssim k$ will 
          be denoted by~$\square^k$\nindex{$g}{$\square^k$}{}. 
          We then write~$\Lambda^k$\nindex{Lambda}{$\Lambda^k$}{} 
          for the set of all open sets 
          $V \subset \square^k$ (in the subspace topology) 
          containing~$\ssim k$.
    \item A map $f \colon \ssim k \longrightarrow M$ is
          \defin{smooth}\dindex{smooth}  
          if there is a $V \in \Lambda^k$ and a smooth map $ F \in \maps{V}{M}$ 
          such that 
          \[ F | _{\ssim k} = f.\]
          The pair~$(F,V)$ is called an 
          \defin{extension of}~$f$.\dindex{extension}
    \item A map $f \colon M \longrightarrow \ssim k$ is called
          \defin{smooth}\dindex{smooth} 
          if the composition $i \circ f$ is smooth, where $i$ is an affine 
          isomorphism $\square^k \longrightarrow \R^k$ composed with the 
          inclusion $\ssim k \hookrightarrow \square^k$. 
          We also use this terminology if $M$ happens to be 
          a standard simplex and $f$ is smooth in the above sense.
    \item We write $\maps[norm]{\ssim k} M$ and $\maps[norm]M {\ssim k}$ for the 
          set of all smooth maps~$\ssim k \longrightarrow M$ and the 
          set of all smooth maps~$M \longrightarrow \ssim k$, respectively.%
          \nindex{mapinfd}{$\maps[norm]{\ssim k} M$}{}% 
          \nindex{mapinfd}{$\maps[norm]M {\ssim k}$}{}%
          \qedhere 
  \end{itemize}
\end{defi}

\begin{bem}\label{smchincl}
  If $M$ is a smooth manifold without boundary 
  and $k\in \N$ then we obviously get an inclusion 
  $\maps[norm]{\ssim k}{M} \hookrightarrow \ssing k M.$
\end{bem}

The mapping space~$\map[norm]{\ssim k} M$ can be topologised as
follows: 

\begin{defi}
  Let $M$ be a smooth manifold without boundary or a standard simplex,
  and let $k \in \N$. We use the notation  
  $\mapsco[norm]{\ssim k}{M}$\dindex{topology!on maps3@on $\mapsco[norm]{\ssim
  k}M$}\nindex{mapco}{$\mapsco[norm]{\ssim k} M$}{} for the 
  set $\maps[norm]{\ssim k} M \subset \map[norm]{\ssim k}M$
  endowed with the subspace topology.
\end{defi}

However, we will also need some kind of \cone-topology on~$\maps[norm]{\ssim k} M$. 
So we introduce the differential of a smooth map $\ssim k \longrightarrow M$:

\begin{defi}
  For $k \in \N$ we use the notation\nindex{TDelta}{$T\ssim k$}{$= \ssim k
  \times \R^k$}
  \[ T\ssim k := \ssim k \times \R^k.\]
  Moreover, if $V \in \Lambda_k$, we will always presuppose the canonical 
  identification 
  \[ TV = V \times \R^k \]
  of the tangent bundle of the smooth $k$-manifold~$V$.
\end{defi}

\begin{defi}\label{defdiff}
  Let $M$ be a smooth manifold without boundary or a standard simplex, 
  and let $k\in \N$. 
  For a smooth map~$f \in \maps[norm]{\ssim k}M$ we define its 
  \defin{differential}\dindex{differential} by
  \begin{align*}
    Tf := TF|_{T\ssim k} \colon T\ssim k \longrightarrow TM
  \end{align*}
  where $(F,V)$ is some extension of~$f$.
\end{defi}

This differential is indeed well-defined:

\begin{lem}
  Let $M$ be a smooth manifold, $k \in \N$, and $f \in \maps[norm]{\ssim k}M.$ 
  Suppose $(F_1,V_1)$ and~$(F_2,V_2)$ are two extensions of~$f$. Then
  \[   TF_1 |_{\pi_1^{-1}(\ssim k)}
     = TF_2 |_{\pi_2^{-1}(\ssim k)} \]
  where $\pi_1 \colon TV_1 \longrightarrow V_1$ and 
  $\pi_2 \colon TV_2 \longrightarrow V_2$ are the bundle projections.
\end{lem}
\begin{bew}
  Since $F_1$ and~$F_2$ are smooth, their 
  differentials $TF_1$ and~$TF_2$ are continuous. Furthermore,
  \[   TF_1 |_{\pi_1^{-1}((\ssim k)^\circ)}
     = TF_2 |_{\pi_2^{-1}((\ssim k)^\circ)} \]
  since $F_1$ and~$F_2$ extend~$f$. For each
  $(x,v) \in \pi_1^{-1}(\ssim k) = \pi_2^{-1}(\ssim k)$ there
  is a sequence $(x_n)_{n \in \N} \subset (\ssim k)^\circ$ such that
  \[ x = \lim_{n \rightarrow \infty} x_n, \]
  which implies $\lim_{n \rightarrow \infty} (x_n, v) = (x,v)$
  in~$\pi_1^{-1}(\ssim k) = \pi_2^{-1}(\ssim k)$.
  Hence we conclude
  \begin{alignat*}{2}
    TF_1(x,v) & = TF_1\bigl(\lim_{n \rightarrow \infty} (x_n, v)\bigr) 
             && = \lim_{n \rightarrow \infty} TF_1(x_n, v) \\
              & = \lim_{n \rightarrow \infty} TF_2(x_n, v) 
             && = TF_2\bigl(\lim_{n \rightarrow \infty} (x_n, v)\bigr)\\
              & = TF_2(x,v). 
             && \qedhere
  \end{alignat*}
\end{bew}

Having defined the differential of a smooth map~$\ssim k \longrightarrow M$, 
we are now able to introduce the \cone-topology 
on~$\maps[norm]{\ssim k} M$:

\begin{defi}
  Let $M$ be a smooth manifold without boundary or a standard simplex,
  and let $k \in \N$. The 
  \cone\defin{-topology on}~$\maps[norm]{\ssim
  k}M$\dindex{c1topology@\cone-topology}\dindex{topology!on
  maps4@on~$\maps[norm]{\ssim k}M$} is the topology induced 
  by the compact open topology on~$\map[norm]{T\ssim k}{TM}$ pulled back 
  by the differential (as defined in Definition~\ref{defdiff})
  \begin{align*}
    T \colon  \maps[big]{\ssim k}M 
       & \longrightarrow \map[big]{T \ssim k}{TM} \\
     f & \longmapsto     Tf. \qedhere
  \end{align*}
\end{defi}

\begin{lem}\label{simfiner}
  Let $M$ be a smooth manifold without boundary or a standard simplex,
  and let $k \in \N$. The topology 
  of~$\maps[norm]{\ssim k}{M}$ is finer than the topology 
  of~$\mapsco[norm]{\ssim k}M$.
\end{lem}
\begin{bew}
  We can use literally the same proof as in Lemma~\ref{finer}.  
\end{bew}

\begin{bem}\label{smmapssim}
  In the same way we can derive statements corresponding to the second 
  part of Lemma~\ref{comp}.
\end{bem}

The following two rather technical results are needed for the proof
that measure homology is compatible with colimits. 

\begin{lem}\label{openincl}
  Let $M$ be a smooth manifold without boundary and $N\subset M$ an
  open subset. Let $k \in \N$.
  \begin{enumerate}
  \item Then the map
        \[ j \colon \maps[big]{\ssim k}{N} 
                    \longrightarrow 
                    \maps[big]{\ssim k}{M}, \]
        induced by the inclusion $N \hookrightarrow M$, is continuous, 
        injective, and open. 
  \item Let $\im j := j\bigl( \maps[norm]{\ssim k}{N}\bigr) \subset 
        \maps[norm]{\ssim k}M$ (endowed with the subspace topology). 
        Then the inverse
        $ \im j \longrightarrow \maps[norm]{\ssim k}N$
        of~$j$ is continuous.
 \end{enumerate}
\end{lem}
\begin{bew}
1. Since the inclusion $N \hookrightarrow M$ is smooth, continuity 
  of~$j$ follows from Remark~\ref{smmapssim}. 
  The fact that $j$ is injective is clear because $N$ is Borel in~$M$.

  The map~$j$ is open: Namely, since $j$ is injective, taking the image of~$j$ is
  compatible with intersections. 
  Hence it suffices to show that 
  $j$ maps sets of the form $T^{-1}(U^K)$ with open $U \subset TN$
  and compact $K \subset T\ssim k$ to open sets in~$\maps[norm]{\ssim k} M$. 
  By definition, 
  \begin{align*}
    & j\bigl( T^{-1}(U^K) \bigr) \\
    & \quad = j \bigl(\bigl\{ f \in \maps[norm]{\ssim k}N
                     \bigm|  Tf(K) \subset U
                     \bigr\} 
                \bigr)\\
    & \quad = \bigl\{ i \circ f 
              \bigm|  \text{$f \in \maps[norm]{\ssim k}N$, $Tf(K) \subset U$}
              \bigr\} \\
    & \quad = \bigl\{ f \in\maps[norm]{\ssim k} M
              \bigm|  Tf(K) \subset U
              \bigr\}
              \cap 
              \bigl\{ f \in \map[norm]{\ssim k} M 
              \bigm|  f(\ssim k) \subset N
              \bigr\} 
  \end{align*}
  where $i \colon N \hookrightarrow M$ is the inclusion. 
  Since $N$ is open in $M$, the set $U$ must be open in $TM$. Thus the first 
  set is open in~$\maps[norm]{\ssim k}M$. Lemma~\ref{simfiner} shows that the 
  second set is also open in~$\maps[norm]{\ssim k}M$. Thus~$j$ is open. 

2. This is an immediate consequence of the first part.
\end{bew}

\begin{koro}\label{nested}
  Let $M$ be a smooth manifold without boundary and let $(U_k)_{k\in \N}$ 
  be an ascending family of open sets covering~$M$. 
  Let $j \in \N$. For 
  each $k \in \N$ the map $\maps[norm]{\ssim j}{U_k} \longrightarrow 
  \maps[norm]{\ssim j}M$ induced by the inclusion $U_k \hookrightarrow M$ 
  is denoted by~$j_k$. Then 
  $\bigl(j_k(\maps[norm]{\ssim j}{U_k})\bigr)_{k \in \N}$
  is an ascending open covering of~$\maps[norm]{\ssim j}M$.
\end{koro}
\begin{bew} 
  It is clear that this family is ascending. 
  By Lemma~\ref{openincl} all the images~$j_k\bigl(\maps[norm]{\ssim j}
  {U_k}\bigr)$ are open in~$\maps[norm]{\ssim j} M$. 


  Moreover, $\maps[norm]{\ssim j}{M}$ is covered by
  $\bigl(j_k(\maps[norm]{\ssim j}{U_k})\bigr)_{k \in \N}$: namely, if
  $f \in \maps[norm]{\ssim j}M$, then the image $f(\ssim j) \subset M$
  is compact. Since $(U_k)_{k \in \N}$ is an ascending open covering
  of~$M$, there is some $k \in \N$ such that $f(\ssim j) \subset U_k$.
  If $(F,V)$ is an extension of~$f$, then $V' := F^{-1}(U_k)$ is an
  open subset of~$V$ containing~$\ssim j$, and the pair~$\bigl(
  F|_{V'}, V' \bigr)$ shows that there is an $\overline f \in
  \maps[norm]{\ssim j}{U_k}$ satisfying
  \[ f = j_k (\overline f). \qedhere\]  
\end{bew}

% -------------------------------------------------------------------
% -------------------------------------------------------------------
\section{Definition of measure homology}\label{mhdefsec}
\iindex{measure homology}

Measure homology is a generalisation of singular homology in the
following sense: Let $X$ be a topological space, $k \in\N$, and let $S
\subset \map{\ssim k} X$ be some set of simplices. The idea of measure
homology is to think of a singular chain~$\fsum_{\sigma \in S}
a_\sigma \cdot \sigma$ with real coefficients as a signed measure on
$S$ having the mass $a_\sigma$ on the set $\{\sigma\}$.  The measure
chain complex will consist of \emph{all} signed measures on~$S$
(modulo some finiteness condition). In particular, certain
``infinite'' singular chains are allowed.  Thus the measure chain
complex is larger than the singular chain complex and hence gives more
room for certain constructions such as smearing (see
Section~\ref{smearing}). The other side of the coin is that it is quite
hard to get a geometric intuition of more complicated measure chains. 

Depending on the choice of the mapping space~$S$ (and its topology) there are 
two main flavours of measure homology:
\begin{itemize}
  \item One for general (pairs of) spaces using the compact open topology on 
        the set of all singular simplices, and
  \item one for (pairs of) smooth manifolds using the \cone-topology on the 
        set of smooth singular simplices.
\end{itemize}
However, our application of measure homology in Chapter~\ref{chpropprinc} 
compels us to use the smooth version. So we will only introduce this version 
in detail.

Measure homology was introduced by Thurston \cite[page~6.6]{thurston}. 
Some basic properties of smooth measure homology are provided in 
Ratcliffe's book \cite[\S11.5]{ratcliffe}.  A thorough treatment of measure
homology for general spaces is given by Zastrow and Hansen
\cite{zastrow}, \cite{hansen}.  In Theorem~3.4 of Zastrow's paper a
way is described how to derive the corresponding results and proofs for
the smooth theory. Just as Zastrow, we refer to those adapted statements
of his paper by numbers such as, e.g., $4.5^\text{diff}$ instead
of~$4.5$.
Unfortunately, it is probably more transparent to provide a new proof
for the compatibility with colimits instead of adapting Hansen's
result \cite[Proposition~5.1]{hansen}. Therefore we give a full proof
at the end of this chapter.

\begin{defi}
  Let $M$ be a smooth manifold without boundary and $k \in \N$. 
  \begin{itemize}
    \item The $k$\defin{-th measure chain group},  denoted by 
          $\mhch{k}{M}$, \nindex{Cm}{$\mhch * M$}{measure chains} 
          is the $\R$-vector space of all signed measures on the Borel
          $\sigma$-algebra of the mapping 
          space~$\maps[norm]{\ssim k}{M}$ possessing a compact
          determination set  
          and finite total variation. Its elements are called
          \defin{measure} $k$\defin{-chains}\dindex{measure
          chain}\dindex{chain!measure}.
    \item For each $j \in \{0, \dots, k+1\}$ the inclusion 
          $\bou_j \colon \ssim k \longrightarrow \ssim{k+1}$ 
          of the $j$-th face induces 
          by Remark~\ref{smmapssim} and Lemma~\ref{mhabschluss} 
          a homomorphism\nindex{dj}{$\bou_j$}{homorphism induced by the $j$-th
          face map} (which we will also denote by~$\bou_j$)
          \begin{align*} 
              \bou_j  \colon \mhch{k+1} M & \longrightarrow \mhch k M\\
                                      \mu & \longmapsto
                \mu^{(\sigma \mapsto \face j \sigma)}
          \end{align*}
  \end{itemize}
          We then define the \defin{boundary operator}\dindex{boundary
          operator}\dindex{measure homology!boundary operator of} of measure
          chains by\nindex{d}{$\bou$}{boundary operator}
          \begin{align*}
            \bou \colon \mhch{k+1} M & \longrightarrow \mhch k M \\
                                 \mu & \longmapsto     
                                       \sum_{j = 0}^{k+1} (-1)^j 
                                             \cdot \bou_j(\mu).
            \qedhere
          \end{align*}
\end{defi}

Without the condition on the variation, we would not get a vector space 
(since signed measures cannot be added in general). The compact 
determination sets play in some way the r\^ole of the compactness of 
the standard simplex (as can be seen in Lemma~\ref{dirlimlem}) and the
finiteness of singular chains in singular homology.

Just as in singular homology theory, we suppress the dependency of~$\bou$ 
on~$k$ in the notation.

\begin{lem}
  Let $M$ be a smooth manifold without boundary and let $k \in \N.$ 
  Then $\bigl( \mhch * M, \bou \bigr)$ is a chain complex.
\end{lem}
\begin{bew}
  This is shown in \cite[Corollary~$2.9^\text{diff}$]{zastrow} and 
  \cite[Lemma~1 in~\S11.5]{ratcliffe}.
\end{bew}

Statements like the previous lemma are proved by Zastrow, using a 
more general framework which permits to translate some properties 
of singular homology to the setting of measure homology 
\cite[Theorem~2.1$^{\text{diff}}$]{zastrow}.

Due to the above lemma, we can naturally define a homology theory:

\begin{defi}
  Let $M$ be a smooth manifold without boundary and $k\in\N$. 
  The $\R$-vector space\nindex{Hm}{$\hmh * M$}{measure homology of~$M$}
  \[ \hmh k M :=     \frac{\ker \bigl( \bou \colon \mhch k M \longrightarrow 
                                                    \mhch{k-1} M
                                \bigr)}
                          {\im  \bigl( \bou \colon \mhch{k+1} M \longrightarrow
                                                    \mhch k M
                                \bigr)}
  \]
  is called the $k$\defin{-th measure homology group of}~$M$\dindex{measure
  homology}\dindex{homology!measure}.
\end{defi}

In view of the goal to compute the simplicial volume via measure
homology, we introduce a norm on measure homology using the total
variation:
  
\begin{defi}
  Let $M$ be a smooth manifold without boundary and let $k\in \N$. 
  The total variation $\tov{\,\cdot\,}$ 
  induces a seminorm\dindex{seminorm!measure homology} on $\hmh k M$ as follows: 
  For all~$\mu \in \mhch k M$ we define
  \nindex{$l}{$\mhnorm{\,\cdot\,}$}{seminorm for measure chains}%
  \[ \mhnorm \mu := \inf\bigl\{ \tov \nu
                        \bigm|  \text{$\nu \in \mhch k M$, 
                                      $\bou(\nu) = 0$, 
                                      $[\nu] = \mu$}
                        \bigr\}. \qedhere
  \]
\end{defi}

Just like in singular homology, there is also a notion of relative
homology groups:

\begin{defi}
  Let $M$ be a smooth manifold without boundary and 
  let $N$ be a smooth submanifold 
  without boundary. Due to Remark~\ref{smmapssim} and 
  Lemma~\ref{mhabschluss}, the inclusion 
  $N \hookrightarrow M$ induces a 
  homomorphism $\mhch * N \longrightarrow 
  \mhch * M$, which is injective since $N$ is measurable. 
  Hence for $k \in \N$ we can define the 
  $k$\defin{-th relative measure chain group}\dindex{relative measure chain
  group}\dindex{measure chain!relative} as the quotient vector
  space\nindex{Cmrel}{$\mhch * {M,N}$}{relative measure chain group}%
  \[ \mhch k {M,N} := \frac{\mhch k M}{\mhch k N}. \qedhere\]
\end{defi}

The boundary operator on the absolute measure chain groups clearly induces a 
boundary operator on the relative measure chain groups, turning them into a
chain complex.

\begin{defi}
  Let $M$ be a smooth manifold without boundary and let $N \subset M$ 
  be a smooth 
  submanifold without boundary. For each $k \in \N$ the $k$\defin{-th relative 
  measure homology group}\dindex{relative measure homology}\dindex{measure
  homology!relative} is defined to be the quotient\nindex{Hmrel}{$\hmh
  *{M,N}$}{relative measure homology group}
  \[ \hmh k {M,N} := 
         \frac{\ker \bigl( \bou \colon \mhch k {M,N} \longrightarrow
                                       \mhch{k-1}{M,N}
                    \bigr)}
              {\im  \bigl( \bou \colon \mhch{k+1}{M,N} \longrightarrow
                                       \mhch k {M,N}
                    \bigr)}. \qedhere
  \]
\end{defi}

\begin{bem}
  Let $M$ be a smooth manifold without boundary. Since there is only one 
  measure on the empty set (the zero-measure), we get 
  $\mhch * \emptyset = 0$. This yields canonical isomorphisms
  \begin{align*}
     \mhch * M & \cong \mhch * {M,\emptyset}, \\
     \hmh  * M & \cong \hmh  * {M,\emptyset}. \qedhere
  \end{align*}
\end{bem}


% --------------------------------------------------------
% --------------------------------------------------------
\section{Basic properties of measure homology}\label{mhprop}

In the category of smooth manifolds without boundary and smooth maps
measure homology behaves like singular homology with real
coefficients; actually, most of the proofs are based on the same ideas
as the corresponding proofs for singular homology theory.

\begin{lem}[Dimension Axiom]
  \tindex{dimension axiom!measure homology}%
  \tindex{measure homology!dimension axiom}%
  Measure homology satisfies the dimension axiom: For all~$n \in \N$
  \[      \hmh n \opsp 
    \cong \begin{cases}
            \,\R & \text{if } n = 0,\\
            \,0 & \text{otherwise.}
          \end{cases}\]
\end{lem}
\begin{bew}
  It is easy to see that the chain complexes $\bigl(\mhch * \opsp, \bou\bigr)$ 
  and~$\bigl(\csingc * \opsp \R, \bou\bigr)$ are the same 
  because for each $k\in\N$ there is just one map $\ssim k \longrightarrow 
  \opsp$ (which is smooth). 
  Thus 
  \[ \hmh * \opsp \cong \hsingc * \opsp \R\]
  which proves the lemma.
\end{bew}

\begin{lem}[Naturality]%
 \tindex{naturality!of measure homology}%
 \tindex{measure homology!naturality of}%
 \nindex{Hm}{$\hmh * f$}{homomorphism induced by~$f$}%
 Each smooth map $f \colon M \longrightarrow N$ between smooth
 manifolds without boundary induces a homomorphism
 \[ \hmh * f \colon \hmh * M \longrightarrow \hmh * N \]
 in measure homology. This turns ${\cal H_*}$ into a functor.
\end{lem}
\begin{bew}
  Since $f$ is smooth, we get for each $k \in \N$ an induced map 
  \begin{align*} 
     \maps[big]{\ssim k}M & \longrightarrow \maps[big]{\ssim k}N \\
                   \sigma & \longmapsto     f \circ \sigma,  
  \end{align*}
  which is continuous by Remark~\ref{smmapssim}. 
  Hence Lemma~\ref{mhabschluss} shows that we obtain an induced map 
  \begin{align*} 
    \mhch k M & \longrightarrow \mhch k N \\ 
          \mu & \longmapsto     \mu^{(\sigma \mapsto f \circ \sigma)}
  \end{align*}
  which obviously is a homomorphism. 
  This yields a chain map 
  $\mhch * M \longrightarrow 
  \mhch * N$ \cite[Lemma~$2.10^{\text{\normalfont{diff}}}$]{zastrow}. 
  Hence we get an induced homomorphism $\hmh * M 
  \longrightarrow \hmh * N$. 
  It is clear that this assignment is functorial.
\end{bew}

\begin{satz}[Long Exact Pair Sequence]%
  \tindex{long exact pair sequence for measure homology}%
  \tindex{measure homology!long exact pair sequence} 
  There is an exact pair sequence for measure homology, i.e., if $M$ is a 
  smooth manifold without boundary and $N \subset M$ is a smooth
  submanifold without boundary,   
  then the sequence 
  \begin{align*}
    \xymatrix@=2.1em{
        {\cdots}          \ar[r] 
      & {\hmh {k+1}{M,N}} \ar[r]
      & {\hmh k N}        \ar[r]^-{\hmh k i}
      & {\hmh k M}        \ar[r]^-{\hmh k j}
      & {\hmh k {M,N}}    \ar[r]
      & {\cdots}
    }
  \end{align*}
  is exact for all $k \in \N$. Here $i \colon N \hookrightarrow M$ and 
  $j \colon (M,\emptyset) \hookrightarrow (M,N)$ are the inclusions.
  The so-called \defin{connecting homomorphism}\dindex{connecting
  homomorphism for measure homology}\dindex{measure homology!connecting
  homomorphism} $\hmh{*+1}{M,N} \longrightarrow 
  \hmh * N$ is natural in $M$~and~$N$.
\end{satz}
\begin{bew}
  The definition of relative measure chain groups yields a short exact sequence
  \begin{align*}
    \xymatrix@=2em{
         {0}             \ar[r] 
       & {\mhch k N}     \ar[r] 
       & {\mhch k M}     \ar[r]
       & {\mhch k {M,N}} \ar[r]
       & {0}},
  \end{align*}
  where the maps are induced by the (smooth) inclusions $i$~and~$j$. 
  Therefore we obtain the desired long exact homology sequence 
  \cite[Theorem~1.31]{weibel}.
\end{bew}

\begin{satz}[Homotopy Invariance]\label{mhhomeq}
  \tindex{homotopy invariance!of measure homology}%
  \tindex{measure homology!homotopy invariance of}%
  Measure homology is a homotopy invariant functor, i.e., if $f$, $g \colon M 
  \longrightarrow N$ are homotopic smooth maps of smooth manifolds without 
  boundary, then
  \[ \hmh * f  = \hmh * g \colon \hmh * M \longrightarrow \hmh * N.\]
\end{satz}

A proof is given in \cite[$5.3^{\text{\normalfont{diff}}}$]{zastrow}.

\begin{bem}
  One might suspect that it is necessary for $f$ and $g$ in the above theorem 
  to be \emph{smoothly} homotopic. But smooth homotopic maps are always
  smoothly homotopic \cite[Proposition 10.22]{leesmooth}.
\end{bem}

Using a barycentric subdivision process, Zastrow proves the following 
theorem \cite[Theorem~$4.1^{\text{\normalfont{diff}}}$]{zastrow}:

\begin{satz}[Excision]\label{mhexc}%
  \tindex{excision!for measure homology}%
  \tindex{measure homology!excision for}%
  Measure homology satisfies the following variation of the excision 
  axiom:
  Let $M$ be a smooth manifold without boundary and let $W \subset M$ be a 
  smooth submanifold without boundary. If $U \subset W$ is a subset such 
  that $M \setminus U$ is a smooth manifold without boundary and 
  $W \setminus U \subset M \setminus U$ is a smooth submanifold without 
  boundary and if there is some subset~$V \subset M$ such that
  \[ \overline U \subset \overset \circ V 
                 \subset \overline V
                 \subset W, \]
  then the inclusion $ (M\setminus U,W \setminus U) \hookrightarrow (M,W)$ 
  induces an isomorphism
  \[ \hmh * {M\setminus U, W \setminus U} \cong 
     \hmh * {M,W}.\] 
\end{satz}


\begin{defi}
  A topological space $X$ is called \defin{normal}\dindex{normal} if disjoint closed
  sets $A$,~$B \subset X$ can be separated by open sets, i.e., there
  are disjoint open sets $U$,~$V \subset X$ with $A \subset U$ and~$B
  \subset V$.
\end{defi}

\begin{bem}\label{mfnormal}%
  All metric spaces are normal. In particular, every smooth manifold is
  normal (since every smooth manifold can be given a Riemannian
  structure, and thus yielding a metric).
\end{bem}


\begin{koro}\label{mhexcsp}
  Let $M$ be a smooth manifold without boundary and $W \subset M$ an 
  open subset. If $U \subset M$ is a closed subset with $U \subset W$, 
  then the inclusion $(M \setminus U, W \setminus U) \hookrightarrow (M,W)$
  induces an isomorphism
  \[ \hmh * {M \setminus U, W \setminus U}\cong \hmh * {M,W}.\]
\end{koro}
\begin{bew}
   Since $W \subset M$ is open, $W$ is a smooth submanifold without boundary. 
   Moreover, $M \setminus U$ is a smooth manifold without boundary and
   the subset $W \setminus U \subset M \setminus U$ is a smooth submanifold without 
   boundary because $U$ is closed.

   Thus it remains to show that there is an open subset~$V \subset M$ with 
   \[ U \subset V \subset \overline V \subset W. \]
   Since $M$ is normal and $U \subset W$, we can separate the disjoint 
   closed sets $U$ and~$M \setminus W$ by open sets $V$ and $V'$. 
   In particular,  
   \[ U \subset V \subset \overline V \subset M \setminus V' 
        \subset W. \]
   Hence excision proves the corollary.    
\end{bew}

\begin{koro}[Mayer-Vietoris Sequence]\label{mhmv}
  \tindex{Mayer-Vietoris sequence for measure homology}%
  \tindex{measure homology!Mayer-Vietoris sequence}%
  Suppose $M$ is a smooth manifold without boundary 
  and $U$, $V\subset M$ are open subsets with
  \[ U \cup V = M.\]  
  Then the sequence 
  \begin{align*}
  \makebox[0pt][c]{%
     \xymatrix@=1.5em{
          {\cdots} \ar[r]
        & {\hmh {k+1}{U \cap V}} 
          \ar[r]^-A
        & {\hmh{k+1} U \oplus \hmh{k+1} V}  
          \ar[r]^-B
        & {\hmh{k+1} M}
          \ar[r]^-{\Delta}
        & {\hmh k {U \cap V}} \ar[r]
        & {\cdots}
     } 
     }
  \end{align*}
  is exact for all $k \in \N$, where 
  \begin{alignat*}{2}
           j_U \colon & U \cap V \hookrightarrow U, 
  & \qquad i_U \colon & U \hookrightarrow M, \\
           j_V \colon & U \cap V \hookrightarrow V, 
  & \qquad i_V \colon & V \hookrightarrow M\phantom{,}
  \end{alignat*}
  are the inclusion maps,
  \begin{align*}
     A & := \bigl(\hmh{k+1}{j_U},-\hmh{k+1}{j_V}\bigr), \\  
     B & := \bigl(\hmh{k+1}{i_U}, \hmh{k+1}{i_V}\bigr),
  \end{align*} 
  and $\Delta$ is defined below.
\end{koro}
\begin{bew}
  By Corollary~\ref{mhexcsp} the inclusion $(V, U\cap V) \hookrightarrow
  (M,U)$ induces an isomorphism $\hmh * {V,U\cap V} \cong
  \hmh * {M,U}.$ We define~$\Delta$ by the composition 
  \begin{align*}
    \xymatrix@=1.4em{
        {\hmh{k+1} M}           \ar[r]
      & {\hmh{k+1}{M,U}} 
      & {\hmh{k+1}{V, U\cap V}} \ar[r] \ar[l]_-{\cong}
      & {\hmh k {U \cap V}}
    }
  \end{align*}
  where the arrow on the left is induced by the inclusion $M 
  \hookrightarrow (M,U)$ and the arrow on the right is the 
  connecting homomorphism of the long exact measure homology 
  sequence of the pair~$(V,U\cap V)$.

  The long exact measure homology sequences of the pairs 
  $(V,U\cap V)$ and $(M,U)$ yield a commutative ladder
  \begin{align*}
    \xymatrix@=1.1em{
        {\cdots}                    \ar[r] 
      & {\hmh {k+1} {U\cap V}}      \ar[r] \ar[d]
      & {\hmh {k+1} V}              \ar[r] \ar[d]
      & {\hmh {k+1} {V,U\cap V}}    \ar[r] \ar[d]_-{\cong}
      & {\hmh {k}   {U\cap V}}      \ar[r] \ar[d]
      & {\cdots} \\
      % -------------------------------------------
        {\cdots}                    \ar[r]
      & {\hmh {k+1} U}              \ar[r]  
      & {\hmh {k+1} M}              \ar[r]
          \ar     `d[r] `[r]+/r1em/_{\Delta} 
                   `_u[ur]+/r1em/+/u1em/|!{[r];[r]}\hole |!{[ur];[ur]}\hole
                   `[urr] `[urr] [urr]  
      & {\hmh {k+1} {M,U}}          \ar[r]
      & {\hmh {k}   U}              \ar[r]
      & {\cdots}
    }
  \end{align*}
  with exact rows. Thus the sequence 
  in the theorem is exact \cite[Lemma~(8.3) in~Kapitel~IV]{td}. 
\end{bew}

Moreover, measure homology is compatible with colimits.

\begin{satz}\label{dirlim}
  \tindex{measure homology!colimits and}%
  Let $M$ be a smooth manifold without boundary covered by an
  ascending  
  family~$(U_k)_{k\in \N}$ of open subsets. Then the inclusions 
  $U_k \hookrightarrow M$ induce an isomorphism 
  \[       \dirlim_{k \rightarrow \infty} \hmh * {U_k} 
     \cong \hmh * M.\]
\end{satz}

The idea of the proof is essentially the same as in singular homology: 
to use the compactness of the standard simplex.  
However, we have to be careful about some technicalities.
The key observation is the following:

\begin{lem}\label{dirlimlem}
  Let $M$ and $(U_k)_{k\in\N}$ be as in the theorem.
  Suppose $j \in \N$ and $\mu \in \mhch j M$. 
  Then we can find an index $k \in \N$ and a measure 
  chain $\nu \in \mhch j {U_k}$ such that
  \[ \mu = \nu^{j_k} \qquad\text{and}\qquad 
     \bou \mu = (\bou \nu)^{j_k}, \] 
  where $j_k \colon \maps[norm]{\ssim j} {U_k} \longrightarrow 
  \maps[norm]{\ssim j}M$ 
  is the map induced by the inclusion~$U_k \hookrightarrow M$. 
  Furthermore, if $\mu$ is a cycle, then so is~$\nu$. 
\end{lem}
\begin{bew}[Proof (of Lemma~\ref{dirlimlem}).]
 By Corollary~\ref{nested} the family $(\im j_k)_{k \in \N}$ is an 
 open ascending covering of~$\maps[norm]{\ssim j}M$, 
 using the notation
 \[ \fa{k \in \N} \im j_k := j_k\bigl( \maps[norm]{\ssim j}{U_k}
                                \bigr).
 \]
 Let $D$ be a compact determination set for~$\mu$. Hence there is an 
 index~$k \in \N$ such that $D \subset j_k\bigl( \maps[norm]{\ssim j}{U_k} 
 \bigr)$. 
 
 Let $\siga_k$ be the Borel~$\sigma$-algebra of~$\im j_k$ (with 
 the subspace topology). By Lemma~\ref{openincl} the 
 set~$\im j_k$ is open in~$\maps[norm]{\ssim j}{M}$. Thus 
 $\siga_k$ is contained in the Borel~$\sigma$-algebra 
 of~$\maps[norm]{\ssim j}{M}$. Therefore we may define the signed 
 measure $\ucov\mu$ on~$\im j_k$ via
 \[ \fa{A \in \siga_k} \ucov\mu(A) := \mu(A). \]
 By construction, $D \subset \im j_k$, implying that $D$ is a 
 compact determination set of~$\ucov\mu$. Using Lemma~\ref{tov},  
 it is easy to see that $\ucov\mu$ has finite total variation. 

 If $J_k \colon \im j_k \longrightarrow \maps[norm]{\ssim j}{U_k}$ is the 
 inverse of~$j_k$, then $J_k$ is continuous by Lemma~\ref{openincl}. 
 Hence the push-forward
 \[ \nu := {\ucov\mu}^{J_k} \]
 is a signed measure on~$\maps[norm]{\ssim j}{U_k}$ 
 having a compact determination set and finite 
 total variation (Lemma~\ref{mhabschluss}), i.e., 
 $\nu \in \mhch j {U_k}$. 

 Our next step is to show the equality~$\mu = \nu^{j_k}$: 
 Let $A \subset \maps[norm]{\ssim j}M$ be some Borel set. 
 Hence 
 \[ \ucov A := A \cap \im j_k \in \siga_k \]
 and 
 \[ \mu(A) = \mu(\ucov A)
           + \mu\bigl( A \cap (\maps[norm]{\ssim j}M \setminus \im j_k)
                \bigr). \]
 Since $\maps[norm]{\ssim j}M \setminus \im j_k$ is a measurable 
 set contained in the complement of the determination set~$D$, 
 \[ \mu(A) = \mu(\ucov A) \]
 holds. Therefore the constructions of $j_k$~and~$J_k$ yield
 \begin{alignat*}{2}
   \mu(A)  & = \mu(\ucov A) 
          && = \ucov\mu(\ucov A) \\
           & = \ucov\mu\bigl( J_k^{-1} \bigl(j_k^{-1}(\ucov A)
                                           \bigr)
                           \bigr) 
          && = {\ucov\mu}^{J_k}
               \bigl( j_k^{-1}(A \cap \im j_k)
               \bigr) \\
           & = \nu
               \bigl( j_k^{-1}(A \cap \im j_k)
               \bigr)
          && = \nu
               \bigl(j_k^{-1}(A)
               \bigr) \\              
           & = \nu^{j_k}(A).
 \end{alignat*}

 Since $j_k$ is induced by the (smooth) inclusion $U_k \hookrightarrow
 M$, the map ${\,\cdot\,}^{j_k}$ is a chain map. 
 In particular,
 \[ \bou\mu = \bou(\nu^{j_k}) = (\bou \nu)^{j_k}.\]

 Suppose $\mu$ is a cycle and $A \subset \maps[norm]{\ssim j}{U_k}$ 
 is a Borel set. Then $j_k(A) = J_k^{-1}(A)$ lies in $\siga_k$. 
 As mentioned above, this results in $j_k(A)$ being Borel 
 in~$\maps[norm]{\ssim j}M$.
 Using the injectivity of~$j_k$ we therefore compute 
 \begin{alignat*}{2}
   \bou\nu(A)   =\bou\nu\bigl(j_k^{-1}(j_k (A)) 
                        \bigr) 
                = (\bou \nu)^{j_k}\bigl( j_k(A)\bigr) 
                = \bou\mu\bigl( j_k(A)\bigr)
                = 0.
 \end{alignat*}
 Hence $\nu$ is also a cycle.
\end{bew}


\begin{bew}[Proof (of Theorem~\ref{dirlim}).]
  For each $k \in \N$ and each $\ell \in \N_{\geq k}$ we write
  \[ i_k \colon U_k \hookrightarrow M \quad\text{and}\quad
     i_k^\ell \colon U_k \hookrightarrow U_\ell\]
  for the inclusions.
  Let $j \in \N$. 
  Then $\bigl( \hmh j{U_k}\bigr)_{k \in \N}$ becomes a direct system 
  with respect to the structure maps 
  $\bigl( \hmh[norm]j{i_k^\ell}\bigr)_{(k,\ell) \in D}$ 
  where 
  \[D := \{ (k,\ell) \in \N^2 | \ell \geq k \}.\] 
  Hence we can form the direct limit $\dirlim_{k \rightarrow \infty}
  \hmh j {U_k}$ of this direct system. 

  The universal property of the direct limit provides us with 
  a homomorphism 
  \[ h_j \colon \dirlim_{k \rightarrow \infty} \hmh j{U_k} 
         \longrightarrow \hmh j M \]
  induced by the maps $\bigl( \hmh j {i_k}\bigr)_{k\in\N}$. 
  To see why~$h_j$ happens to be an isomorphism we use the following
  fact \cite[Corollary~D.3]{bredon}: it is sufficient to show that for
  each~$\alpha \in \hmh j M$ there exists a~$k \in \N$ and a~$\beta
  \in \hmh j{U_k}$ such that $\hmh j {i_k}{\beta} = \alpha$, and that for
  each $\alpha \in \hmh j {U_k}$ with $\hmh j {i_k}(\alpha) = 0$ there
  is an $\ell \in \N$ such that $\hmh j {i_k^\ell}(\alpha) = 0$.
  \begin{enumerate}
  \item%
    Let $\mu \in \mhch j {M}$ be a measure cycle. Then the 
    preceding lemma provides us with some $k \in \N$ and a measure 
    cycle~$\nu \in \mhch j {U_k}$ such that $\mu = \nu^{j_k}$. This 
    proves
    \[ \hmh j{i_k} \bigl( [\nu]\bigr) = [\mu] \]
    in $\hmh j M$, since $j_k$ is induced by the inclusion~$i_k$. 
  
  \item%
    Let $k \in \N$ and $\nu \in \mhch j{U_k}$ be a measure cycle such 
    that $\hmh j {i_k}\bigl([\nu]\bigr) = 0 \in \hmh j M$. 
    Hence there exists a measure chain $\tau \in \mhch{j+1}M$ with 
    \[ \bou \tau = \nu^{j_k}. \]
    The above lemma  gives us an $\ell \in \N$ (and the proof shows that 
    we can assume~$\ell \geq k$) and a measure chain~$\varrho \in 
    \mhch{j+1}{U_\ell}$ satisfying 
    \begin{align*}
      \varrho^{j_\ell}        & = \tau, \\
      (\bou \varrho)^{j_\ell} & = \bou \tau = \nu^{j_k}.
    \end{align*}
    Writing $j_k^\ell \colon \maps[norm]{\ssim j}{U_k} 
    \longrightarrow \maps[norm]{\ssim j}{U_\ell}$ for the map induced 
    by the inclusion $i_k^\ell$, we conclude 
    \[   (\bou \varrho)^{j_\ell} 
       = \nu^{j_k} 
       = \bigl(\nu^{j_k^\ell}\bigr)^{j_\ell}. \]
    As in the last part of the proof of Lemma~\ref{dirlimlem}, we can 
    compute for all Borel sets~$A \subset \maps[norm]{\ssim j}{U_k}$
    \begin{alignat*}{2}
      \bou\varrho(A) 
         & = \bou\varrho\bigl(j_\ell^{-1}(j_\ell(A))
                        \bigr) 
        && = (\bou\varrho)^{j_\ell}\bigl( j_\ell(A)\bigr) \\
         & = \bigl(\nu^{j_k^\ell}\bigr)^{j_\ell}
             \bigl(j_\ell(A)\bigr)
        && = \nu^{j_k^\ell}\bigl( j_\ell^{-1}(j_\ell(A))\bigr) \\
         & = \nu^{j_k^\ell}(A).
    \end{alignat*}
    This shows $\hmh j {i_k^\ell}\bigl([\nu]\bigr) = 0$ in~$\hmh j
    {U_\ell}$. \qedhere
  \end{enumerate}
\end{bew}

With the help of these algebraic properties we will see in
Section~\ref{algisom} that measure homology and singular homology
(algebraically) coincide.


% -------------------------------------------------------------------
% -------------------------------------------------------------------
%
\chapter{Measure Homology is Singular Homology}\label{chisomisom}

\begin{chapterabstract}
In this chapter, singular homology (with real coefficients) and
measure homology of smooth manifolds without boundary are compared.
The results from Section~\ref{mhprop} already suggest that the
homology groups of these two theories should coincide. Surprisingly,
the natural isomorphism providing this correspondence turns out to be
\emph{isometric}. It is thus possible to calculate the simplicial
volume of an oriented  smooth closed connected manifold via measure homology.

In order to obtain the mentioned isometric isomorphism between singular homology and 
measure homology, we proceed in three steps:
\begin{itemize}
  \item First of all, we introduce smooth singular homology (for smooth
        manifolds without boundary) and show that it is 
        canonically isometrically isomorphic to singular homology.
  \item In the second step, we establish an (algebraic) isomorphism
        between smooth singular homology and measure homology.
        This is done by means of the properties described in
        Section~\ref{mhprop}. 
  \item Thirdly, we prove that this isomorphism is actually isometric. This 
        is the central contribution of this thesis. The method of
        proof is inspired by the duality principle of bounded
        cohomology (Theorem~\ref{duality}).
\end{itemize}
In the last section, we discuss an integration operation for measure
chains as a tool to analyse the top homology group, which will turn
out to be useful in the context of smearing (Section~\ref{smearing}).
\end{chapterabstract}



\section{Smooth singular homology}

Since measure homology is based on smooth simplices, it is easier to
see a connection between measure homology and smooth singular homology
instead of singular homology.
Smooth singular homology is defined just like ordinary singular
homology, using smooth singular chains instead of continuous ones. 

\begin{defi}
  Let $M$ be a smooth manifold without boundary and $k\in \N$. The
  elements of $\maps[norm]{\ssim k} M$ are called \defin{smooth singular}
  $k$-\defin{simplices of}~$M$.\dindex{smooth singular
  simplex}\dindex{simplex!smooth singular}
  The subcomplex (using the inclusion from Remark~\ref{smchincl})
  \nindex{Csm}{$\smsingc * M \R$}{set of smooth singular simplices in~$M$}%
  \nindex{d}{$\bou$}{boundary operator}
  \[ \bigl( \smsingc * M \R, \bou|_{\smsingc * M \R} \bigr) \subset 
     \bigl( \csingc   * M \R, \bou \bigr) \]
  generated by all smooth singular simplices yields the \defin{smooth singular 
  homology groups}\dindex{smooth singular homology}\dindex{homology!smooth
  singular}\nindex{Hnsm}{$\hsmsingc * M \R$}{smooth singular homology of~$M$}
  \[ \hsmsingc k M \R 
      :=     \frac{\ker \bigl( \bou%|_{\smsingc k M \R}
                               \colon \smsingc k M \R \longrightarrow 
                                      \smsingc{k-1} M \R
                        \bigr)}
                  {\im  \bigl( \bou%|_{\smsingc{k+1} M \R}
                               \colon \smsingc{k+1} M \R \longrightarrow
                                      \smsingc k M \R
                        \bigr)}
  \]
  for each $k \in \N$.
\end{defi}

More details about smooth singular homology can be found in the
textbooks of Massey and Lee \cite[\S2 in Appendix~A]{massey},
\cite[Chapter~16]{leesmooth}.  In Massey's book it is shown (using the
same methods as in ordinary singular homology) that smooth singular
homology satisfies theorems corresponding to those stated in
Section~\ref{mhprop}.  These statements will in the following be
referred to via numbers such as~$\text{\ref{mhexc}}^\text{sm}$ instead
of~\ref{mhexc}.

Moreover, by the same pattern as for singular homology, we can define a 
norm~$\smlone{\,\cdot\,}$ on the smooth singular chain complex and its
induced seminorm $\smlone{\,\cdot\,}$ on smooth singular homology.
\nindex{$k}{$\smlone{\,\cdot\,}$}{seminorm on smooth singular homology}%
\dindex{seminorm!smooth singular homology}

\begin{satz}\label{singsmooth}\tindex{isometric isomorphism}
  For every smooth manifold $M$ without boundary the natural chain complex
  inclusion 
  \nindex{jM}{$j_M$}{inclusion $\smsingc{*}{M}{\R} \hookrightarrow
  \csingc{*}{M}{\R}$\\} 
  \[ j_M \colon \smsingc{*}{M}{\R} \hookrightarrow \csingc{*}{M}{\R}\]
  induces an isometric isomorphism between the homology
  groups $\hsmsingc{*}{M}{\R}$ and $\hsingc{*}{M}{\R}$.
\end{satz}
\begin{bew}
  Via the Whitney approximation theorem, a smoothing operator
  \nindex{s}{$s$}{smoothing operator}%
  \[ s \colon \csingc{*}{M}{\R} \longrightarrow \smsingc{*}{M}{\R}\]
  can be constructed \cite[page~417]{leesmooth}, satisfying
  the following conditions:
  \begin{itemize}
    \item The map $s$ is a chain map with
          \[ s \circ j_M = \id \quad\text{and}\quad j_M \circ s \simeq \id.\]
    \item For each singular simplex $\sigma \in \ssing k M$ the 
          image~$s(\sigma) \in \smsingc k M \R$ consists of just one 
          smooth simplex. 
  \end{itemize}
  The first part implies that $\hsing{*}{j_M} \colon \hsmsingc * M \R 
  \longrightarrow \hsingc * M \R$ is an isomorphism. Its inverse
  is~$H_*(s)$. 

  From the second part we deduce: if $\fsum_{\sigma \in \ssing k M}
  a_\sigma \cdot \sigma \in \csingc k M \R$  is a cycle representing the 
  homology class~$\alpha \in \hsingc k M \R$, then
  $\fsum_{\sigma \in \ssing k M} a_\sigma \cdot s(\sigma)$ is a cycle 
  representing $\hsing k s (\alpha)$ with
  \begin{align*}
    \smlone{\hsing k s (\alpha)} 
  & \leq \smlone[bigg]{\fsum_{\sigma \in \ssing k M} a_\sigma \cdot s(\sigma)}
    \leq \fsum_{\sigma \in \ssing k M} |a_\sigma| \\
  & =    \lone[bigg]{ \fsum_{\sigma \in \ssing k M} a_\sigma \cdot \sigma}.
  \end{align*}
  Taking the infimum over all representatives of~$\alpha$ thus yields 
  \[ \smlone{\hsing k s (\alpha)} \leq \lone \alpha 
                             =    \lone{\hsing k {j_M}
                                  \bigl( \hsing k s(\alpha)\bigr)}.\]
  On the other hand, obviously
  \[ \lone{\hsing k {j_M} (\alpha')} \leq \smlone{\alpha'}\]
  holds for all $\alpha' \in \hsmsingc k M \R$.
  Since $\hsing k s$ is a bijection this proves the theorem.
\end{bew}

In particular, we detect the fundamental class in smooth singular
homology and we can compute the simplicial volume via this class: 

\begin{defi}
  Let $M$ be an oriented smooth closed connected manifold of dimension~$n$. 
  The above theorem implies that\dindex{fundamental class!in smooth
  singular homology}%
  \nindex{$x}{$\smfcl M$}{fundamental class in smooth singular homology}
  \[ \smfcl M := \bigl(H_n(j_M)\bigr)^{-1}\bigl(\fclr M\bigr)\]
  is a well-defined generator of~$\hsmsingc n M \R$.
\end{defi}

\begin{koro}\label{isomisomsm}
  Let $M$ be an oriented smooth closed connected manifold. Then 
  \[ \smlone[big]{\smfcl M} = \sv M\]
  holds by the above theorem. \hfill$\square$
\end{koro}

In Section~\ref{isometricisom}, we will make use of a version of
bounded cohomology based on smooth singular homology. 
Its construction is completely analogous to the corresponding definition
of bounded (singular) cohomology and it turns out to be the same as
ordinary bounded cohomology. However, bounded smooth cohomology is
more feasible in the context of measure homology.

\begin{defi}
  Let $M$ be a smooth manifold without boundary, and $k \in\N$. 
  \begin{itemize}
    \item The supremum norm\dindex{supremum norm} of~$f \in \Hom_\R
          \bigl(\smsingc k M \R, \R\bigr)$ is given by
          \nindex{$r}{$\supn{\,\cdot\,}$}{supremum norm}%
          \[ \supn f := \sup_{\sigma \in \maps[norm]{\ssim k} M} 
                        |f(\sigma)|.\]
    \item We write\nindex{Cosm}{$\bsmcch * M$}{bounded smooth cochains of~$M$}
          \[ \bsmcch k M := 
             \bigl\{ f \in \Hom_\R \bigl(\smsingc k M \R, \R\bigr)
             \bigm|  \supn f < \infty
             \bigr\}
          \]
          for the group of \defin{bounded smooth}
          $k$\defin{-cochains}.\dindex{bounded smooth
          cochain}\dindex{cochain!bounded smooth}
    \item The corresponding coboundary operator\dindex{coboundary operator} 
          is defined via\nindex{delta}{$\cobou$}{coboundary operator\\}
          \begin{align*}
              \cobou 
              \colon \bsmcch{k}   M & \longrightarrow \bsmcch {k+1} M \\
                                  f & \longmapsto     
                                   \bigl(c \mapsto (-1)^{k+1} \cdot f(\bou(c)) 
                                   \bigr).
          \end{align*}
    \item The $k$\defin{-th bounded smooth cohomology group}\dindex{bounded
          smooth cohomology}\dindex{bounded cohomology!smooth} $\bsmc k M$
          \defin{of}~$M$ is the $k$-th cohomology group
          of the cochain complex~$(\bsmcch * M, \cobou)$.%
          \nindex{Hobsm}{$\bsmc * M$}{bounded smooth cohomology}
    \item The supremum norm
          induces (as in the case of ordinary bounded cohomology) a
          seminorm\dindex{seminorm!smooth bounded
          cohomology}\nindex{$q}{$\supn{\,\cdot\,}$}{seminorm}
          on the quotient~$\bsmc k M$, which will also be denoted 
          by~$\supn{\, \cdot \,}$.
          \qedhere
  \end{itemize}  
\end{defi}

\begin{qedbem}
   As in the case of ordinary bounded cohomology, it is easy to verify that
   the homomorphism~$\cobou$ is well-defined and satisfies $\cobou \circ
   \cobou = 0$.
\end{qedbem}

Using the smoothing operator~$s$ of the proof of
Theorem~\ref{singsmooth}, it can readily be seen that $\bsmc * M$ and
$\bch * M$ are isometrically isomorphic:

\begin{satz}\label{bousmisom}\tindex{isometric isomorphism}
  Let $M$ be a smooth manifold without boundary. Then the restriction map 
  (where $j_M$ is defined as in Theorem~\ref{singsmooth})
  \begin{align*}
    \bchn * {j_M} \colon
    \bchn * M & \longrightarrow \bsmcch * M \\
            f & \longmapsto     f|_{\smsingc * M \R} = f \circ j_M
  \end{align*}
  induces an isometric isomorphism $\bch * {j_M} \colon
  \bch * M  \longrightarrow \bsmc * M$ on cohomology.
\end{satz}
\begin{bew}
  It is clear that the restriction map is a well-defined cochain map. 
  
  Let $h_*\colon \csingc * M \R \longrightarrow \csingc {*+1} M \R$ be a 
  chain homotopy $j_M \circ s \simeq \id$. The same arguments as in the proof 
  of Lemma~\ref{bchomotopyinv}
  yield that the ``dual'' 
  \begin{align*}
     \bchn * M & \longrightarrow \bchn {*-1} M \\
             f & \longmapsto     (-1)^* \cdot f \circ h_{*-1}
  \end{align*}
  is well-defined and a cochain homotopy $\bchn * s \circ \bchn *{j_M} 
  \simeq \id$, where 
  $\bchn * s$ and $\bchn * {j_M}$ 
  are the ``duals'' of $s$ and $j_M$ respectively. 
  Moreover, 
  \[\bchn *{j_M} \circ \bchn *s = \id, \] 
  by functoriality. 
  Hence the restriction map induces an isomorphism on cohomology. 

  Since $s$ maps singular simplices to singular simplices, 
  we obtain 
  \[\supn {f \circ s} \leq \supn f\] 
  for all~$f \in \bsmcch * M$. On the
  other hand, obviously $\supn{f \circ j_M} \leq \supn f$ holds for
  all~$f \in \bchn * M$. 
  Therefore we conclude 
  \[ \fa{\varphi \in \bch * M}
          \supn{\varphi} = \supn[big]{\bch * s
                                 \bigl( \bch *{j_M}  (\varphi) \bigr)}
     \leq \supn[big]{\bch * {j_M}(\varphi)} 
     \leq \supn{\varphi},
  \]
  which implies that $\bch *{j_M}$ is isometric.
\end{bew}

As in the case of ordinary bounded cohomology, we can define a
\defin{Kronecker product}\dindex{Kronecker product!for bounded smooth
  cohomology \hspace*{5pt}}\dindex{bounded smooth cohomology!Kronecker product
  for}\nindex{$d}{$\krp{\,\cdot\,}{\cdot\,}$}{Kronecker product}
\[ \krp{\, \cdot \,}{\cdot \,} 
   \colon
   \bsmcch * M \times \smsingc * M \R 
   \longrightarrow
   \R. \]
by evaluation. Analogously, this product descends to a bilinear map
\[ \krp{\, \cdot \,}{\cdot \,} 
   \colon
   \bsmc * M \times \hsmsingc * M \R 
   \longrightarrow
   \R, \]
and it is readily seen to be compatible with the ordinary Kronecker
product, i.e., for all $\varphi \in \bch * M$ and all $\alpha \in \hsmsingc * M \R$ 
\[   \krp[big]{\bch * {j_M}(\varphi)}{\alpha} 
   = \krp[big]{\varphi}{H_*(j_M)(\alpha)}. \]
We now obtain a smooth version of the duality principle
(compare Theorem~\ref{duality}), which will become useful in 
Section~\ref{isometricisom}.

\begin{koro}[Duality Principle of Bounded Smooth Cohomology]\label{smduality}%
  \tindex{duality principle!of bounded smooth cohomology\\ \hspace*{5pt}}%
  \tindex{bounded smooth cohomology!duality principle of}
  \hfill Let\linebreak 
  $M$ be a smooth manifold without boundary and let $k \in \N$ 
  and $\alpha \in \hsmsingc k M \R$.
  \begin{enumerate}
    \item Then $\smlone \alpha = 0$ if and only if
          \[ \fa{\varphi \in \bsmc k M} \krp \varphi \alpha = 0.\]
    \item If $\smlone \alpha > 0$, then
          \[ \smlone \alpha = \sup \Bigl\{ \frac 1{\supn\varphi} 
                              \Bigm| \varphi \in\bsmc k M, 
                                     \krp{\varphi}{\alpha} = 1
                              \Bigr\}.\]
  \end{enumerate} 
\end{koro}
\begin{bew}
  By the above remark,
  \begin{align*} 
       \krp{\varphi}\alpha 
   & = \krp[big]{\bch k {j_M} \circ \bch k {j_M}^{-1}(\varphi)}{\alpha} \\
   & = \krp[big]{\bch k {j_M}^{-1}(\varphi)}{H_k(j_M)(\alpha)}
  \end{align*}  
  holds for all $\alpha \in \hsmsingc k M \R$ and all~$\varphi \in \bch
  k M$. Since $\widehat H^k(j_M)$ and $H_k(j_M)$ are
  isometric isomorphisms by Theorem~\ref{bousmisom} and
  Theorem~\ref{singsmooth}, the duality principle~\ref{duality} 
  applied to $H_k(j_M)(\alpha) \in \hsingc k M \R$ proves
  the corollary.
\end{bew}


\section{The algebraic isomorphism}\label{algisom}

In this section, smooth singular homology (with real coefficients)
and measure homology are shown to be isomorphic as vector spaces. 
This isomorphism is induced by a natural chain map:

\begin{lem}\label{inclsmmh}
  Let $M$ be a smooth manifold without boundary. Then there
  is a norm-preserving injective chain map
  \nindex{iM}{$i_M$}{inclusion $\smsingc * M \R \rightarrow \mhch * M$\\}
  \[ i_M \colon \smsingc * M \R \longrightarrow \mhch * M.\]
\end{lem}
\begin{bew}
  Let $k\in\N$.
  For $\sigma \in \maps[norm]{\ssim k}{M}$ we denote the atomic
  measure on~$\maps[norm]{\ssim k}{M}$ concentrated in~$\sigma$
  by~$\delta_\sigma$, which is obviously an element of~$\mhch k M$. 
  If $c : = \fsum_{\sigma \in \maps[norm]{\ssim k} M}
  a_\sigma \cdot \sigma$ is a smooth chain, we define
  \[ i_M(c) := \fsum_{\sigma \in \maps[norm]{\ssim k} M} 
                     a_\sigma \cdot \delta_\sigma \in \mhch k M.\]
  The calculation showing that $i_M$ is a chain map is quite
  straightforward \cite[Lemma~2 in~\S11.5]{ratcliffe}. 

  It remains to show that $i_M$ is norm-preserving. 
  With the above notation we obtain that
  the Hahn decomposition of~$i_M(c)$ is given by 
  \begin{align*}
    P & := \bigl\{ \sigma \in \maps[norm]{\ssim k} M 
           \bigm|  a_\sigma \geq 0
           \bigr\}, \\
    N & := \bigl\{ \sigma \in \maps[norm]{\ssim k} M
           \bigm|  a_\sigma < 0
           \bigr\}.
  \end{align*}
  Hence we conclude 
  \begin{align*}
    \tov{i_M(c)} & = |i_M(c)| \bigl( \maps[norm]{\ssim k} M \bigr) \\
                 & = i_M(c)(P) - i_M(c)(N) \\
                 & = \fsum_{\sigma \in P} a_\sigma
                   - \fsum_{\sigma \in N} a_\sigma \\
                 & = \fsum_{\sigma \in \maps[norm]{\ssim k} M} |a_\sigma| \\
                 & = \smlone c. \qedhere
  \end{align*}
\end{bew}

\begin{satz}\label{smshmh}
  If $M$ is a smooth manifold without boundary, then the inclusion 
  $i_M \colon \smsingc * M \R \hookrightarrow \mhch * M$ of the previous 
  lemma induces an isomorphism
  \[ \hsmsingc * M \R \cong \hmh * M.\]
\end{satz}

In the case of measure homology based on the compact open topology
(instead of the $\cone$-topology), proofs of this theorem were given by
Hansen and Zastrow via the verification of the Eilenberg-Steenrod
axioms \cite[Theorem~1.1]{hansen}, \cite[Theorem~3.4]{zastrow}.
In the smooth setting however, it is not entirely clear how to use
this approach since only smooth manifolds can be handled (instead of
the whole universe of \cw-complexes). 

Therefore we give a proof based on some ``manifold induction'' -- following
a general idea of Milnor: 

\begin{satz}\label{mfdind}
 Assume $P$ is some property  
 of smooth manifolds
 without boundary satisfying the following conditions:
 \begin{enumerate}
  \item The property $P$ is stable under smooth homotopy equivalences.
  \item The property $P$ holds for the one point space (which can be
        thought of as a zero-dimensional smooth manifold without
        boundary). 
  \item If $U$, $V$ are open subsets of the same smooth manifold 
        without boundary such that $U$,~$V$ and~$U\cap V$ have 
        property~$P$, then the union~$U\cup V$ also has property~$P$.
  \item If $(U_k)_{k \in \N}$ is an ascending sequence of subsets of
        some (common) smooth manifold without boundary all having  
        property~$P$, then $\bigcup_{k \in \N} U_k$ satisfies~$P$. 
 \end{enumerate}
 Then each smooth manifold without boundary has property~$P$.
\end{satz}
\begin{bew}
The (not very difficult) proof can for example be found in Massey's
book \cite[Case~5 on page~364]{massey}.
\end{bew}



\begin{bew}[Proof (of Theorem~\ref{smshmh}).]
  We consider the property~$P$ which is defined as follows: 
  a smooth manifold $U$ without boundary satisfies~$P$ if
  and only if the homomorphism
  \[ H_*(i_U)\colon \hsmsingc * U \R \longrightarrow \hmh * U \]
  is an isomorphism. 

  In view of Theorem~\ref{mfdind} it suffices to show that $P$ has the
  four mentioned properties: 

  1. Suppose $f \colon M \longrightarrow N$ is a smooth homotopy 
     equivalence of smooth manifolds without boundary where $M$ 
     enjoys~$P$. Then the diagram 
     \begin{align*}
       \xymatrix@=3em{
         {\hsmsingc * M \R \ar[r]^-{\hsmsingc * f \R}}  \ar[d]_{H_*(i_M)}
       & {\hsmsingc * N \R}                             \ar[d]^{H_*(i_N)} \\
       % ----------------------------------------
         {\hmh * M}         \ar[r]_-{\hmh * f} 
       & {\hmh * N}
       }
     \end{align*}
     is commutative and all arrows but the right vertical one are isomorphisms. 
     Hence~$H_*(i_N)$ must also be an isomorphism, showing 
     that $N$ fulfils~$P$.

  2. The chain complexes $\mhch * \opsp$ and $\smsingc * \opsp \R$ 
     are identical and the inclusion $i_\opsp \colon \mhch * \opsp 
     \hookrightarrow \smsingc * \opsp \R$ is the identity. Hence the 
     one point space satisfies~$P$.

  3. Let $U$, $V$ be open subsets of some smooth manifold $M$ without 
     boundary such that $U$, $V$, and $U\cap V$ satisfy~$P$. Then we 
     obtain the following commutative ladder
     \begin{footnotesize}
     \newcommand{\versum}[2]{%
       *\txt{%
             $#1$ \\ $\oplus$ \\ $#2$%
            }%
     }%
     \newcommand{\sversum}[2]{%
        \txt{%
             \scriptsize $#1$ \\ \scriptsize $\oplus #2$%
            }%
     }%
     \begin{align*}
       \makebox[0pt][c]{%
       \xymatrix@C=1.6em@R=3.4em{
         {\hsmsingc{k+1}{U \cap V}\R}   \ar[r] \ar[d]_{H_{k+1}(i_{U\cap V})}^\cong
       & {\versum{\hsmsingc{k+1}U \R}%
                 {\hsmsingc{k+1}V \R}}%
         \ar[r] %
         \ar[d]_{%
                 \sversum{H_{k+1}(i_u)}%
                         {H_{k+1}(i_V)}%
                }^\cong
       & {\hsmsingc{k+1}{U \cup V} \R}  \ar[r] \ar[d]_{H_{k+1}(i_{U\cup V})}
       & {\hsmsingc k {U \cap V} \R}    \ar[r] \ar[d]_{H_k(i_{U\cap V})}^\cong
       & {\versum{\hsmsingc k U \R}%
                 {\hsmsingc k V \R}}%
         \ar[d]_{H_k(i_U) \oplus H_k(i_V)}^\cong \\
         % ---------------------------------------
         {\hmh{k+1}{U \cap V}}          \ar[r] 
       & {\versum{\hmh{k+1} U}%
                 {\hmh{k+1} V}} \ar[r]
       & {\hmh{k+1}{U\cup V}}           \ar[r] 
       & {\hmh{k}{U \cap V}}            \ar[r] 
       & {\versum{\hmh k U}%
                 {\hmh k V}} 
       }}%
     \end{align*}%
     \end{footnotesize}%
     whose rows are Mayer-Vietoris sequences and thus exact 
     (compare Corollary~\ref{mhmv} and \ref{mhmv}$^{\text{sm}}$). 
     Now the five lemma implies that the 
     union~$U \cup V$ has property~$P$.

  4. Let $M$ be a smooth manifold without boundary and $(U_k)_{k \in \N}$ 
     an ascending sequence of open subsets of~$M$. Suppose that each
     $U_k$ satisfies~$P$. 
 
     Since $\bigcup_{k \in \N} U_K$ is a smooth manifold without boundary 
     covered by the ascending family $(U_k)_{k\in \N}$ of open sets, we 
     conclude from Theorem~\ref{dirlim} and Theorem~\ref{dirlim}$^\text{sm}$
     that the inclusions induce isomorphisms
     \begin{align*}
       i \colon \dirlim_{k \rightarrow \infty} \hmh * {U_k} 
         & \longrightarrow \hmh * U, \\
       j \colon \dirlim_{k \rightarrow \infty} \hsmsingc * {U_k} \R 
         & \longrightarrow \hsmsingc * U \R 
     \end{align*}
     using the notation
     \[ U := \bigcup_{k \in \N} U_k.\]
     Furthermore, for each $k\in \N$ the diagram
     \begin{align*}
       \xymatrix@=4.5em{
          {\hsmsingc * {U_k}\R \ar[r]^-{\hsmsingc * {U_k \hookrightarrow U} \R}} 
                              \ar[d]_{H_*(i_{U_k})} 
        & {\hsmsingc * U    \R} \ar[d]^{H_*(i_U)} \\
          {\hmh * {U_k}}        \ar[r]_-{\hmh * {U_k \hookrightarrow U}}
        & {\hmh * U}
        }
     \end{align*}
     commutes. Since each $U_k$ 
     has property~$P$, we conclude 
     that the induced homomorphism 
     \[ \dirlim_{k \rightarrow \infty} H_*(i_{U_k})
        \colon 
        \dirlim_{k \rightarrow \infty} \hsmsingc * {U_k} \R
        \longrightarrow
        \dirlim_{k \rightarrow \infty} \hmh * {U_k}
     \]
     is an isomorphism.
     The previous diagram shows that
     \begin{align*}
        \xymatrix@=3em{
          {\dirlim_{k \rightarrow \infty} \hsmsingc * {U_k} \R}  
            \ar[r]^-{j} \ar[d]_{\dirlim_{k \rightarrow \infty}H_*(i_{U_k})}
        & {\hsmsingc * U \R  \ar[d]^{H_*(i_U)}} \\
          % ---------------------------------
          {\dirlim_{k \rightarrow \infty} \hmh * {U_k}} 
            \ar[r]_-{i}
        & {\hmh * U}
        }
     \end{align*}
     is commutative, implying that $H_*(i_U)$ is also an isomorphism.
\end{bew}

In particular, we obtain a fundamental class for measure homology. It
is the aim of this chapter to show that this fundamental class can be
used to compute the simplicial volume.

\begin{defi}
  Let $M$ be a smooth oriented closed connected manifold of dimension~$n$.
  By Theorem~\ref{smshmh} the \defin{measure homology fundamental
  class}\dindex{fundamental class!in measure homology}\dindex{measure
  homology!fundamental class}\nindex{$w}{$\mhfcl M$}{measure homology fundamental
  class}
  \[ \mhfcl M := H_n(i_M)\bigl( \smfcl M \bigr) 
     \in \hmh n M\]
  is a generator of 
  \[ \hmh n M \cong \hsmsingc n M \R 
              \cong \hsingc n M \R 
              \cong \R. \qedhere
  \]
\end{defi}


% ---------------------------------------------------------------------
% ---------------------------------------------------------------------

\section{\dots is isometric}\label{isometricisom}\tindex{isometric
  isomorphism}


In this section it will be shown that the (algebraic) isomorphism from
Theorem~\ref{smshmh} is actually isometric.  Thurston already stated
that this should be true \cite[page~6.6]{thurston}, however the paper
cited there has never been published in the suggested form.  In early
2004, a preprint by Bowen appeared to remedy this situation
\cite{bowen}, but the given argument does not seem to be correct (see
Subsection~\ref{bowen}).

We recall the following notation which will be used throughout this
section:

\begin{defi}
  If $M$ is a smooth manifold without boundary, we write
  $j_M \colon \smsingc * M \R \hookrightarrow \csingc * M \R$
  for the obvious inclusion of chain complexes and 
  $i_M \colon \smsingc * M \R \longrightarrow \mhch * M$
  for the injective norm-preserving chain map which is given by mapping 
  smooth singular simplices to the corresponding atomic measure 
  (compare~\ref{inclsmmh}). 
\end{defi}

Then the result can be formulated as follows:

\begin{satz}\label{isomisom}\tindex{isometric isomorphism}
  If $M$ is a smooth manifold without boundary, then the inclusion $i_M 
  \colon \smsingc * M \R \hookrightarrow \mhch * M$ 
  induces an isometric isomorphism
  \[ \hsmsingc * M \R \cong \hmh * M.\]
\end{satz}

Before giving the proof we state the main consequence of this 
theorem:

\begin{qedkoro}\label{isomisomc}%
  \tindex{isometric isomorphism}%
  \tindex{simplicial volume!measure homology and}%
  \tindex{measure homology!simplicial volume and}%
  \tindex{singular homology!measure homology and}%
  \tindex{measure homology!singular homology and}
  Let $M$ be a smooth manifold without boundary. 
  \begin{enumerate}
    \item Then the inclusions~$i_M$ and $j_M$ induce an isometric
          isomorphism 
          \[ \hsingc * M \R \cong \hmh * M. \]
    \item In particular, the simplicial volume of~$M$ can be computed
          via measure homology, i.e.,
          \[ \sv{M} = \mhnorm[big]{\mhfcl M}.\qedhere\]
  \end{enumerate}
\end{qedkoro}

\begin{bem}
  The second part was already known to be true for hyperbolic
  manifolds \cite[page~235]{gromov}, \cite[Exercise~10
  in~\S11.5]{ratcliffe} but the general case was still open.
\end{bem}

\begin{bem}
  It seems possible to use the same proof in the case of measure
  homology for nice enough metric spaces (based on the compact open topology on
  the set of singular simplices), hence giving an isometric
  isomorphism between singular homology and measure homology for
  general (metric) spaces. 
\end{bem}

The proof of the theorem is based on the idea of imitating the duality
results of Theorem~\ref{duality} (and Corollary~\ref{smduality}). 
This yields a reformulation of the problem in terms of bounded
cohomology, which can be solved by means of the established
framework sketched in Section~\ref{bcgroupssection}. 

The drawback of this approach is that the proof of
Theorem~\ref{isomisom} is not as geometric as one might wish.

\subsection{A dual for measure homology}

As a first step we have to construct the ``dual''~$\bmc * M$  
which plays the r\^ole of  
the bounded (smooth) cohomology groups $\bsmc * M$ in the (smooth)
singular theory.  
If $c = \fsum_{\sigma \in \maps[norm]{\ssim k}M} a_\sigma \cdot \sigma \in
\smsingc k M \R$ 
is a smooth singular chain and $f \in \bsmcch k M$ is a singular cochain,
their Kronecker product is given by
\[    \krp{f}{c} = f(c)
    = \fsum_{\sigma \in \maps[norm]{\ssim k} M} a_\sigma \cdot
      f(\sigma)
    \in \R. \]
If we think of $c$ as a linear combination of atomic measures, this
looks like an integration of~$f$ over~$c$. Hence our ``dual'' in
measure homology will consist of (bounded) functions which can be
integrated over measure chains:

\begin{defi}
  Let $M$ be a smooth manifold without boundary and let $k\in\N$. 
  We write 
  \nindex{Com}{$\bmcch k M$}{bounded Borel measurable functions
  $\maps[norm]{\ssim k} M \rightarrow \R$}%
  \[  \bmcch k M
      :=  \bigl\{ f \colon \maps[norm]{\ssim k}{M} \longrightarrow \R
          \bigm | \text{$f$ is Borel measurable and bounded}
          \bigr\} \]
  and 
  \nindex{delta}{$\cobou$}{coboundary operator\\}%
  \dindex{coboundary operator}%
  \begin{align*}
     \cobou \colon \bmcch k M & \longrightarrow  \bmcch {k+1} M\\
                            f & \longmapsto      
                            \bigl( \sigma \mapsto (-1)^{k+1} \cdot f(\bou(\sigma)) 
                            \bigr).
  \end{align*}
  Here $f(\bou(\sigma))$ is an abbreviation for~$\sum_{j =0}^{k+1} 
  (-1)^j \cdot f(\face j \sigma)$.
\end{defi}
The next lemma shows that 
this map $\cobou  \colon \bmcch k M \longrightarrow  \bmcch {k+1} M$
is indeed well-defined and turns $\bmcch * M$ into a cochain complex.

\begin{lem}
  Let $M$ be a smooth manifold without boundary, $k \in \N$, and let $f
  \in \bmcch k M$. Then $\cobou (f) \in \bmcch{k+1} M$ and
  \[ \cobou \circ \cobou (f) = 0.\]
\end{lem}
\begin{bew}
  According to Remark~\ref{smmapssim},
  \begin{align*}
     \maps[norm]{\ssim {k+1}} M & \longrightarrow \maps[norm]{\ssim k}M \\
                         \sigma & \longmapsto     
                                  \face j \sigma
  \end{align*} 
  is continuous for each $j \in \{0, \dots, k + 1 \}$. Hence 
  $\cobou(f)$ is Borel measurable. As $f$ is bounded, 
  $\cobou(f)$ is bounded as well. 

  Since $\bou (\bou(\sigma)) = 0$ holds for all $\sigma \in
  \maps[norm]{\ssim{k+2}} M$, it follows that
  \[ \cobou \circ \cobou(f) = 0. \qedhere\]
\end{bew}

\begin{defi}
  Let $M$ be a smooth manifold without boundary and let $k \in
  \N$. The 
  \dindex{bounded measure cohomology}%
  \dindex{cohomology!bounded measure}% 
  $k$\defin{-th bounded measure cohomology group of}~$M$ is given by
  \nindex{Hobm}{$\bmc * M$}{bounded measure cohomology of~$M$}%
  \[ \bmc k M := H^k \bigl(\bmcch * M, \cobou \bigr).\] 
  We write \nindex{$q}{$\supn{\,\cdot\,}$}{seminorm}%
  $\supn{\,\cdot\,}$ for the seminorm\dindex{seminorm!bounded
  measure cohomology\\ \hspace*{25pt}} on~$\bmc k M$ which is induced
  by the supremum norm on~$\bmcch k M$.
\end{defi}

\begin{defi}
  Let $M$ be a smooth manifold without boundary and let $k \in
  \N$. The \defin{Kronecker product}\dindex{Kronecker product!for
  measure homology}\dindex{measure homology!Kronecker product} 
  of $\mu \in \mhch k M$ and~$f \in \bmcch k M$ is defined as
  \nindex{$d}{$\krp{\,\cdot\,}{\cdot\,}$}{Kronecker product}%
  \[ \krp f \mu := \int f \;d\mu. \]
  If $\mu$ is a measure cycle and $f$ is a cocycle, we write
  \[ \krp[big]{[f]}{[\mu]} := \krp f \mu = \int f \;d\mu. \qedhere\]
\end{defi}

It is easy to see that this is well-defined and -- as desired --
yields a generalisation of the ordinary Kronecker product:


\begin{lem}\label{mhkrp}%
  \tindex{Kronecker product!for measure homology}%
  \tindex{measure homology!Kronecker product}
  Let $M$ be a smooth manifold without boundary and let $k \in \N$.
  \begin{enumerate}
    \item The Kronecker product $\krp{\,\cdot\,}{\cdot\,} \colon
          \bmcch k M \times \mhch k M \longrightarrow \R$ is
          well-defined and bilinear.
    \item The Kronecker product $\krp{\,\cdot\,}{\cdot\,} \colon
          \bmc k M \times \hmh k M \longrightarrow \R$ on (co)homology
          is well-defined and bilinear.
    \item The Kronecker product defined above is compatible with the
          Kronecker product on bounded smooth cohomology 
          in the following sense: 
          for all $ f \in \bmcch k M$ and all $ c \in \smsingc k M \R$,
          \[ \krp {B(f)} c = \krp f {i_M (c)}, \]
          where $B(f)$\nindex{B}{$B$}{} 
          denotes the linear extension of~$f \colon
          \maps[norm]{\ssim k} M \longrightarrow \R$ to the vector 
          space~$\smsingc k M \R.$          
          Passage to (co)homology yields for all $\varphi \in \bmc k
          M$ and all $\alpha \in \hsmsingc k M \R$ \margcomm{notatie?}
          \[ \krp[big]{H^k(B)(\varphi)}{\alpha}
           = \krp[big]{\varphi}{H_k(i_M)(\alpha)}. 
             \qedhere \]
  \end{enumerate}
\end{lem}
\begin{bew}
  1. The integral is defined and finite,  
     since the elements of $\mhch k M$ are (signed)
     measures of finite total variation and the elements of~$\bmcch k
     M$ are bounded measurable functions. 
     Moreover, the integral is obviously bilinear.
   
  2. It remains to show that the definition of the Kronecker product
     on (co)ho\-mol\-ogy 
     does not depend on the chosen representatives. This is a
     consequence of the following fact: if $\ell \in \N$, $\mu \in
     \mhch \ell M$ and~$f \in \bmcch{\ell-1} M$, then the
     transformation formula yields  
     \[ \int_{\maps{\ssim {\ell} } M } f \circ \partial_j \; d\mu  
      = \int_{\maps{\ssim {\ell-1} } M } f \;d\mu^{(\sigma \mapsto \face j \sigma)}\]
     for all $j \in \{0, \dots, \ell\}$, and hence $\int \cobou(f) \;d\mu 
     = \int f \; d\bou(\mu)$.

  3. Since both Kronecker products are bilinear, it suffices to
     consider the case where $c$ consists of a single smooth
     simplex~$\sigma \in \maps[norm]{\ssim k} M$. Then the left hand side -- by
     definition -- evaluates to~$f(\sigma)$. Since $i_M(c)=
     i_M(\sigma) = \delta_\sigma$ is the atomic measure on~$\maps[norm]
     {\ssim k} M$
     concentrated in~$\sigma$, we obtain for the right hand side
     \[ \krp f {i_M(c)} = \int f \;d\delta_\sigma = 1 \cdot f(\sigma).
     \]
     The corresponding equality in (co)homology follows because $B
     \colon \bmcch * M \longrightarrow \bsmcch * M$ is
     easily recognised to be a chain map.
\end{bew}

The above Kronecker product leads to the following (slightly weakened)
duality principle.

\begin{lem}[Duality Principle of Measure Homology]\label{mhduality}%
  \tindex{duality principle!of measure homology}%
  \tindex{measure homology!duality principle of}
  Let $M$ be a smooth manifold without boundary, $k\in\N$, and $\alpha
  \in \hmh k M$.
  \begin{enumerate}
    \item If $\mhnorm \alpha = 0$, then for all $\varphi \in \bmc k M$
          \[ \krp \varphi \alpha = 0. \]
    \item If $\mhnorm \alpha > 0$, then 
          \[ \mhnorm \alpha \geq \sup 
                  \Bigl\{ \frac 1 {\supn \varphi} 
                  \Bigm|  \text{$\varphi \in \bmc k M$, $\krp
                          \varphi \alpha = 1$}
                  \Bigr\}.\]
  \end{enumerate}
\end{lem}
\begin{bew}
  Let $\varphi \in \bmcch k M$. Assume that $\mu \in \mhch k
  M$ is a measure cycle representing $\alpha$ and $f \in \bmcch k M$
  is a cocycle representing $\varphi$.
  \margcomm{spaces!!} 
  If $\krp \varphi \alpha =1$, then
  \begin{align*}
     1 & =    \lvert\krp \varphi \alpha\rvert   
         =    \lvert\krp f \mu\rvert \\
       & =    \Bigl|\int f \; d\mu\Bigr|
         \leq \int |f| \;d\mu_+ + \int |f| \;d\mu_- \\
       & \leq \supn f \cdot \int 1 \;d\mu_+
            + \supn f \cdot \int 1 \;d\mu_- \\
       & =      \supn f \cdot \tov\mu. 
  \end{align*}
  Taking the infimum over all representatives results in
  $1 \leq \supn \varphi \cdot \mhnorm \alpha.$
  In particular, if there exists such a $\varphi$, then
  \[ \mhnorm \alpha \geq \frac 1 {\supn \varphi} > 0.\]
  Now the lemma is an easy consequence of this inequality.
\end{bew}

A posteriori we will be able conclude -- in view of Theorem~\ref{isomisom},
and Lemma~\ref{abde} -- 
that in the first part of the lemma ``if and only if'' is also true
and that in the second part equality holds.

\subsection{Proof of Theorem~\ref{isomisom}}\tindex{isometric isomorphism}

The dual $\bmcch * M$ will be investigated by means of the complex
$\bfch * M$ introduced in Definition~\ref{specialdef}: the vector
space $\bfch k M$ is the set of all bounded functions in~$\map[norm]{\ucov
M^{k+1}}{\R}$ which are $\pi_1(M)$-invariant.  
Then the  key to the proof of Theorem~\ref{isomisom} is a careful analysis of the 
diamond 
\begin{align*}
  \xymatrix@=3em@dr{%
    {\bfch  * M}  \ar^{D}[r] \ar_{A}[d] & {\bchn * M},   \ar^{E}[d]\\
    {\bmcch * M}  \ar_{B}[r]        & {\bsmcch * M} 
  }
\end{align*}
the maps being defined as follows:%
\nindex{A}{$A$}{}%
\nindex{B}{$B$}{}%
\nindex{D}{$D$}{}%
\nindex{E}{$E$}{}
\begin{itemize}
 \item For $k \in \N$ let $s_k \colon \map[norm]{\ssim k} M \longrightarrow
       \map[norm]{\ssim k}{\ucov M}$ be a Borel section of the map induced
       by the universal covering map. The existence of such a section 
       is guaranteed by the following theorem:
       
       \begin{satz}\label{borelsection}
         Let $M$ be a smooth manifold without boundary with universal
         covering map ~$p \colon \ucov M \longrightarrow M$, and let
         $k\in \N$. 
         \begin{enumerate}
            \item Then the map
                  \begin{align*}
                     P \colon \map[big]{\ssim k}{\ucov M} & \longrightarrow
                              \map[big]{\ssim k} M \\
                              \sigma                 & \longmapsto
                              p \circ \sigma
                  \end{align*}
                  is a local homeomorphism.
            \item Moreover, there exists a Borel section of~$P$.
         \end{enumerate}
       \end{satz} 
   
       The (elementary, but rather technical) proof of this theorem is
       exiled to Subsection~\ref{exile}. 

       If $f\in \bfch k M$, we write 
       \begin{align*}
         A(f) \colon \maps[big]{\ssim k} M & \longrightarrow \R \\
                             \sigma   & \longmapsto
                                        f\bigl( (s_k(\sigma))(e_0)
                                               , \dots, 
                                               (s_k(\sigma))(e_k) 
                                         \bigr),
       \end{align*}
       using the inclusion $\maps[norm]{\ssim k}M \hookrightarrow \map[norm]{\ssim k}{M}$.
 \item Similarly (see also Section~\ref{specialres}), we define $D(f)$
       for $f \in \bfch k M$
       as the linear extension to~$\csingc k M \R$ of the map
       \begin{align*}      
         \ssing k M =
         \map[big]{\ssim k} M & \longrightarrow \R \\
                  \sigma & \longmapsto f\bigl( (s_k(\sigma))(e_0)
                                       , \dots, 
                                       (s_k(\sigma))(e_k) 
                                \bigr).
       \end{align*}
 \item The map $B$ is given by linear extension (cf. Lemma~\ref{mhkrp}).
 \item The map $E$ is given by restriction, i.e., $E = \bchn *{j_M}.$
\end{itemize}

This diagram allows us to compare the map~$B$ (which is the 
building bridge between $\bmc * M$ and~$\bsmc * M$) with $D$ and~$E$, 
which both induce isometric isomorphisms on the level of bounded cohomology.


\begin{lem}\label{abde}
  With the above notation the following statements hold:
  \begin{enumerate}
    \item The maps $A$, $B$, $D$, and $E$ are well-defined cochain maps.
    \item Moreover, $\supn{A(f)} \leq \supn f$ for all~$f \in \bfch k
          M$ and all $k \in \N$. This implies
          \[ \fa{\xi \in H^k\left( \bfch * M\right)} 
                \supn[big]{H^*(A) (\xi)} \leq \supn \xi.\] 
    \item The diagram commutes, that is $B \circ A = E \circ D$.
    \item The homomorphism $H^*(B) \colon \bmc * M \longrightarrow \bsmc * M$ 
          induced by the cochain map~$B$ is surjective.
  \end{enumerate}
\end{lem}
\begin{bew}
  1. The fact that $B$ and $E$ are well-defined and compatible with the
     respective coboundary operators can be directly read off the
     definitions.
 
     In Section~\ref{specialres} it is proved that $D$ does not depend on
     the choice of~$s_k$ and that $D$ is a cochain map. The same
     calculations show that the definition of~$A$ is independent of
     the choice of~$s_k$ and that $A$ is a cochain map. Let $f \in \bfch k
     M$. Then $A(f)$ is obviously bounded. Moreover, $A(f)$ is
     measurable: By construction, $A(f)$ is the composition
      \newlength{\pinguin}
      \settowidth{\pinguin}{$(\tau(e_0),  \dots, \tau(e_k))$}
      \begin{alignat*}{4}
        \maps[big]{\ssim k} M       & \hookrightarrow 
        \map[big]{\ssim k} M        & \xrightarrow{s_k} 
        \map[big]{\ssim k}{\ucov M} & \rightarrow       
        \makebox[\pinguin][c]{$\ucov M^{k+1}$}             
        & \xrightarrow{f} 
        \R \\
        \sigma & \mapsto 
        \sigma &
        \tau   & \mapsto
        (\tau(e_0),  \dots, \tau(e_k)) &  
       \end{alignat*}
     which consists of measurable and continuous functions (see 
     Lemma~\ref{simfiner}, Remark~\ref{smmapssim}, Remark~\ref{ev}).     

  2. This is an immediate consequence of the definition of~$A$.

  3. For all $f \in \bfch k M$ and all $\sigma \in \maps[norm]{\ssim
     k}M$ we obtain 
     \begin{align*}
           \bigl( B \circ A(f) \bigr)(\sigma) 
       & = \bigl( A(f) \bigr) (\sigma) \\
       & = f\bigl( (s_k(\sigma))(e_0)
                 , \dots
                 , (s_k(\sigma))(e_k) \bigr) \\
       & = \bigl(D(f)\bigr)\bigl(j_M(\sigma)\bigr) \\
       & = \bigl( E \circ D(f)\bigr)(\sigma).
     \end{align*}

  4. Due to Corollary~\ref{specialcor} and Theorem~\ref{bousmisom},  
     the chain maps $D$ and $E$ induce isomorphisms on cohomology. Since the 
     above diagram commutes, $H^*(B)$ must be surjective.
\end{bew}


We have now collected all the necessary tools to prove that measure homology 
and (smooth) singular homology are isometrically isomorphic.

\begin{bew}[Proof (of Theorem \ref{isomisom}).]
  According to Theorem~\ref{smshmh}, 
  the induced homomorphism $H_*(i_M) \colon \hsmsingc * M 
  \R \longrightarrow \hmh * M$ is an isomorphism. Therefore, it remains 
  to show that~$H_*(i_M)$ is compatible with the seminorms.

  Let $k \in \N$ and $\alpha \in \hsmsingc k M \R$. Since $i_M 
  \colon \smsingc * M \R \longrightarrow \mhch * M$ is norm preserving, 
  it is immediate that
  \[ \mhnorm{H_k(i_M)(\alpha)} \leq \smlone{\alpha}. \]
  The proof of the reverse inequality is split into two cases:

\ding{192} Suppose $ \mhnorm[big]{H_k(i_M)(\alpha)} = 0$.  From 
  Lemma~\ref{mhkrp} and Lemma~\ref{mhduality} we obtain
  \[ \krp[big]{H^k(B)(\varphi)}{\alpha} 
   = \krp[big]{\varphi}{H_k(i_M)(\alpha)} 
   = 0\]
  for all $\varphi \in \bmc k M$.  By the previous lemma, $\bch k B$ 
  is surjective. Hence
  \[ \fa{\psi \in \bsmc k M} \krp \psi \alpha = 0,\]
  implying $\smlone \alpha = 0$.

\ding{193} Let $\mhnorm[big]{H_k(i_M)(\alpha)} > 0$. 
  In this case, Lemma~\ref{mhduality} and Lemma~\ref{mhkrp} yield
  \begin{align*}
    \mhnorm[big]{H_k(i_M)(\alpha)} 
  & \geq \sup
    \Bigl\{ \frac 1 {\supn \varphi} 
    \Bigm| \text{$\varphi \in \bmc k M$,  $\krp[big] \varphi {H_k(i_M)(\alpha)} =1$}
    \Bigr\} \\
  & = \sup
    \Bigl\{ \frac 1 {\supn \varphi} 
    \Bigm| \text{$\varphi \in \bmc k M$,  $\krp[big]{H^k(B) (\varphi)} \alpha =1$}
    \Bigr\}.
  \end{align*}
  We will compare the last set with the corresponding set of 
  Corollary~\ref{smduality}:
  Let $\psi \in \bsmc k M$ such that $\krp \psi \alpha = 1$. Since
  the composition $H^k(E) \circ H^k(D)$ is an isometric
  isomorphism (Corollary~\ref{specialcor} and Theorem~\ref{bousmisom}), 
  there exists a $\xi \in H^k\bigl( \bfch * M \bigr)$
  satisfying 
  \[ \bigl( H^k(E) \circ H^k(D) \bigr) (\xi) = \psi 
     \quad\text{and}\quad \supn \xi = \supn \psi.\]
  Then $\varphi := H^k(A)(\xi) \in \bmc k M$ possesses the following properties:
  \begin{itemize}
    \item By construction,
          \[ H^k(B) (\varphi) 
           = \bigl(H^k(B) \circ H^k(A)\bigr)(\xi)
           = \bigl(H^k(E) \circ H^k(D)\bigr)(\xi) 
           = \psi,\]
          and hence 
          \[ \krp[big]{H^k(B) (\varphi)}{\alpha} 
           = \krp{\psi}{\alpha} 
           = 1.\]
    \item Furthermore, we get from Lemma~\ref{abde}
          \[ \supn \varphi  = \supn[big]{H^k(A)(\xi)}
             \leq \supn \xi = \supn \psi.\]
  \end{itemize}      
  Combining these properties with the above estimate results in
  \begin{align*} 
    \mhnorm[big]{H_k(i_M)(\alpha)} 
  & \geq \sup
    \Bigl\{ \frac 1 {\supn \varphi} 
    \Bigm| \text{$\varphi \in \bmc k M$,  $\krp[big]{H^k(B)(\varphi)} \alpha =1$}
    \Bigr\} \\
  & \geq \sup
    \Bigl\{ \frac 1 {\supn \psi}
    \Bigm| \text{$\psi \in \bsmc k M$, $\krp{\psi}{\alpha} = 1$}
    \Bigr\}.
  \end{align*}
  Since $\smlone \alpha \geq \mhnorm{H_k(i_M)(\alpha)} > 0$, we can
  use Lemma~\ref{smduality} to conclude 
  \[\mhnorm{H_k(i_M)(\alpha)} \geq \smlone \alpha.\qedhere\]
\end{bew}

\subsection{Existence of a Borel section}\label{exile}

To complete the proof of Theorem~\ref{isomisom}, we still have to
provide a proof of Theorem~\ref{borelsection}:

\begin{bew}[Proof (of Theorem~\ref{borelsection}).]
  Since $M$ is smooth, we may assume that $\ucov M$ is also a smooth manifold and 
  $p$ is a local diffeomorphism. Moreover, the smooth manifold~$M$ can
  be equipped with a Riemannian metric. Then there is a Riemannian
  metric on~$\ucov M$ such that $p$ is a local isometry (e.g., the
  pulled back Riemannian metric). The
  topologies on $M$ and $\ucov M$ are induced by the metrics $d_M$ and
  $d_{\ucov M}$\nindex{dM}{$d_M$}{metric on~$M$}\nindex{dMtilde}{$d_{\ucov
  M}$}{metric on~$\ucov M$} 
  which are given by the respective Riemannian
  metrics. Then $p$ is also a local isometry
  with respect to~$d_M$ and $d_{\ucov M}$. 

  1. Let $\sigma\in \map[norm]{\ssim k}{\ucov M}$. Since $p \colon
  \ucov M \longrightarrow M$ is a covering map, there is a small
  neighbourhood~$U$ of~$\sigma(e_0)$ in~$\ucov M$, which means:

  \begin{defi}\dindex{small}
    An open subset $U \subset \ucov M$ is called \defin{small} if
    $p(U) \subset M$ is open and $p|_U \colon U \longrightarrow p(U)$
    is a homeomorphism.
  \end{defi}

  We will show that $P(U^{\{ e_0\}})$ is open and that 
  $P|_{U^{\{ e_0\}}} \colon U^{\{e_0\}} \longrightarrow P(U^{\{e_0\}})$
  is a homeomorphism.

  \emph{The set $P(U^{\{ e_0\}})$ is open in~$\map[norm]{\ssim k} M$.} By
  definition, $P(U^{\{ e_0\}}) \subset \bigl(p(U)\bigr)^{\{e_0\}}$. On
  the other hand, for each $\tau \in \bigl(p(U)\bigr)^{\{e_0\}}$ there
  exists a lift $\ucov \tau \colon \ssim k \longrightarrow \ucov M$
  such that $\ucov \tau(e_0) \in U$ since $\ssim k$ is simply
  connected. 
  Thus
  \[ P(U^{\{ e_0\}}) = \bigl(p(U)\bigr)^{\{e_0\}}.\]
  Since $p(U)$ is open in $M$, this is an open subset of~$\map[norm]{\ssim k}
  M$. 

  \emph{The restriction $P|_{U^{\{e_0\}}} \colon U^{\{e_0\}}
  \longrightarrow \bigl( p(U)\bigr)^{\{e_0\}} =
  P\bigl(U^{\{e_0\}}\bigr)$ is bijective.}
  Since $U$ is small, $p|_U$ is injective. Hence the uniqueness of
  lifts (prescribed on $e_0$ by the property to map into $U$) shows 
  injectivity ($\ssim k$ is connected).

  \emph{The restriction $P|_{U^{\{e_0\}}} \colon U^{\{e_0\}}
  \longrightarrow \bigl( p(U)\bigr)^{\{e_0\}} =
  P\bigl(U^{\{e_0\}}\bigr)$ is a homeomorphism.} 
  Because of Lemma~\ref{comp}, the map $P$ is continuous. It therefore
  remains to prove that the restriction $P|_{U^{\{e_0\}}}$ is open: 

  Let $\rho \in U^{\{e_0\}}$. Since $\ssim k$ is compact and $p \circ
  \rho$ is continuous, the set
  \[ K := p \circ \rho (\ssim k)\]
  is compact.

  Then the ``distance'' of the sheets of~$p$ over~$K$ is uniformly bounded
  from below in the following sense:

  \begin{lem}\label{covdist}
    There is a constant $C\in \R_{> 0}$ satisfying the following
    property: the distance of all points~$y_1, y_2 \in p^{-1}(K)$ such that $p(y_1) =
    p(y_2)$ and $y_1 \neq y_2$ is bounded from
    below via
    \[ d_{\ucov M}(y_1,y_2)  \geq C.\]
  \end{lem}
  \begin{bew}[Proof (of Lemma~\ref{covdist}).]
    We consider the function
    \begin{align*}
       D \colon K & \longrightarrow \R \\
                x & \longmapsto     \inf_{y \in p^{-1}(x)
                                          \setminus\{\ucov x\}}
                                    d_{\ucov M}(\ucov x, y),
    \end{align*}
    where $\ucov x$ is some $p$-lift of~$x$. The function~$D$ is
    independent of this choice since the fundamental group~$\pi_1(M)$
    acts isometrically on~$\ucov M$ and transitively on each fibre
    (Corollary~\ref{quotcomp} and \cite[Lemma~V~8.1]{massey}). 

    Let $x \in K$. Then there is an open neighbourhood~$W$ of~$x$
    such that there is a trivialisation 
    \begin{align*}
      \xymatrix@=2em{%
        {W \times \pi_1(M)} \ar[rr]^{\varphi_W}_{\cong}
                            \ar[rd]_{\text{pr}_W}
     & & {p^{-1}(W)}        \ar[ld]^p \\
        & {W} & 
      }
    \end{align*}
    and $p|_{\varphi_W(W \times \{1\})} \colon \varphi_W(W \times \{ 1\})
    \longrightarrow W$ is an isometric diffeomorphism. 
 
    Let $\ucov x := \varphi_W(x,1)$. Since 
    $W$ is an open neighbourhood of~$x$, the image  
    $\varphi_W(W \times\{1\})$  
    is an open neighbourhood of~$\ucov x$ in~$p^{-1}(W)$ (the
    fibre~$\pi_1(M)$ carries the discrete topology) and hence
    in~$\ucov M$. So there is an~$r_x \in \R_{>0}$ such that the open
    ball~$\ballu{2r_x}{\ucov x}$%
    \nindex{Brx}{$\ball{r}{x}$}{open ball in~$M$ of radius~$r$ around~$x$}%
    \nindex{Brxtilde}{$\ballu{r}{x}$}{open ball in~$\ucov M$ of radius~$r$ around~$x$}
    is contained in~$\varphi_W(W \times
    \{1\})$ and $p\bigl( \ballu{2r_x}{\ucov x}\bigr)= \ball{2r_x} x$.
    
    Let $z$ be a point in the open neighbourhood $V_x := \ball{r_x}x$
    of~$x$. We have  $\ball{r_x}{x} \subset W$ because $p|_{\varphi_W(W
    \times \{1\})}$ is isometric and $\ball{2r_x}{x} =
    p\bigl(\ballu{2r_x}{\ucov x}\bigr)$. 
    Since $\varphi_W$ is injective and $\ballu{2r_x}{\ucov x} \subset
    \varphi_W(W \times \{1\})$, we conclude
    \[ \varphi_W(z,h) \not\in \ballu{2r_x}{\ucov x} \]
    for all $h \in \pi_1(M) \setminus \{1\}$.
    In particular, 
    \begin{align*}
            d_{\ucov M}\bigl( \varphi_W(z,h), \varphi_W(z,1) 
                       \bigr)
     & \geq d_{\ucov M}\bigl( \varphi_W(z,h), \ucov x
                     \bigr)
          - d_{\ucov M}\bigl( \ucov x, \varphi_W(z,1)
                     \bigr) \\
     & \geq 2\cdot r_x - d_M(x,z) \\
     & \geq 2\cdot r_x - r_x \\
     & =    r_x
    \end{align*}
    for all $h \in \pi_1(M) \setminus\{1\}$. This yields
    $D(z) \geq r_x$
    for all $z \in V_x$. Since $K$ is compact, it can be covered by finitely
    many such sets $V_{x_1}, \dots, V_{x_n}$. Then
    \[ C := \min_{j \in \{1, \dots, n\}} r_{x_j}\]
    gives the desired estimate.
  \end{bew}  

  We will show below that for all small enough $\varepsilon \in \R_{>
  0}$ the image of 
  \nindex{Vepsilontilde}{$\vtilepsi$}{}%
  \[ \vtilepsi  := \Bigl\{ \tau \in \map[norm]{\ssim k}{\ucov M} 
                  \Bigm| \sup_{x \in \ssim k} d_{\ucov M}(\tau(x), \rho(x))
                       < \varepsilon
                  \Bigr\} \]
  under $P|_{U^{\{e_0\}}}$ is the set
  \nindex{Vepsilon}{$\vepsi$}{}%
  \[ \vepsi     := \Bigl\{ \tau \in \map[norm]{\ssim k}M
                  \Bigm| \sup_{x \in \ssim k}d_M(\tau(x), p\circ \rho(x))
                       < \varepsilon
                  \Bigr\}. \]
  As $\ssim k$ is compact, the topologies on $\map[norm]{\ssim
  k}{\ucov M}$ and $\map[norm]{\ssim k} M$ are the same as the
  respective 
  topologies of uniform convergence (Remark~\ref{uniform}). 
  This proves that $P|_{U^{\{e_0\}}}$ is open.

  The idea is to apply the above lemma in the following way: since the
  sheets of the covering are far away from each other and
  functions in~$\vepsi$ are very close, their lifts must lie in the ``same''
  sheet. Using the fact that  $p$ is a local isometry, we can conclude that the
  lifts are also very close.

  In order to make the second statement precise, we need the following result:

  \begin{lem}\label{werder}
    There is a constant~$\delta \in \R_{>0}$ such that for each $y \in
    p^{-1}(K)$ the restriction 
    \[ p|_{\ballu \delta y} \colon \ballu \delta y  
                            \longrightarrow 
                            p\bigl( \ballu \delta y\bigr)\]
    to the open ball of radius~$\delta$ around~$y$ (with respect
    to~$d_{\ucov M}$) is an isometric diffeomorphism and 
    $p\bigl( \ballu{\delta}{y}\bigr) = \ball{\delta}{p(y)}$. 
  \end{lem}
  \begin{bew}[Proof (of Lemma~\ref{werder}).]
    For each~$x \in K$ we choose an element $\ucov x \in
    p^{-1}(x)$. Then there are balls~$\ballu{\varepsilon_x}{\ucov x}$ such
    that $p|_{\ballu{\varepsilon_x}{\ucov x}} \colon 
    \ballu{\varepsilon_x}{\ucov x} \longrightarrow p\bigl(
    \ballu{\varepsilon_x}{\ucov x}\bigr)$ is an isometric
    diffeomorphism and $p\bigl(
    \ballu{\varepsilon_x}{\ucov x}\bigr) =  \ball{\varepsilon_x}x$.
 
    Since $\pi_1(M)$ acts by isometries on~$\ucov M$ (see Corollary~\ref{quotcomp}), 
    we conclude that
    \[ g \cdot \ballu{\varepsilon_x}{\ucov x} 
     = \ballu{\varepsilon_x}{g \cdot \ucov x}\]
    and that
    \[ p|_{g \cdot \ballu{\varepsilon_x}{\ucov x}} \colon 
       g \cdot \ballu{\varepsilon_x}{\ucov x} \longrightarrow p\bigl(
       \ballu{\varepsilon_x}{\ucov x}\bigr) \]
    is an isometric diffeomorphism for all~$g \in \pi_1(M)$.
    As $K$ is compact, we can find an~$n \in \N$ and $x_1, \dots, x_n
    \in K$ such that 
    \[ K \subset \bigcup_{j \in \{1,  \dots, n\}}
                 \ball{\varepsilon_{x_j}/2}{x_j}. \]
    Then
    \[ \delta := \min_{j \in \{1, \dots,n\}}
                 \frac{\varepsilon_{x_j}}2\]
    is the desired constant: namely, let $y \in p^{-1}(K)$ and $x :=
    p(y)$. Then there is a~$j \in \{1, \dots, n \}$ such that $x \in
    \ball{\varepsilon_{x_j}/2}{x_j}$. Since $\pi_1(M)$ acts
    transitively on each fibre, there is a~$g \in \pi_1(M)$ satisfying
    $y \in g \cdot \smash{\ballu{\varepsilon_{x_j}}{\ucov{x_j}}}$. 
    Because $\smash{p|_{\ballu{\varepsilon_{x_j}}{g \cdot \ucov{x_j}}}}$ is an
    isometry, 
    \[ d_{\ucov M}(y,  g \cdot \ucov{x_j}) 
     = d_M(x, x_j) \leq \frac{\varepsilon_{x_j}}2.\]
    In particular, $\ballu{\delta}{y} \subset
    \ballu{\varepsilon_{x_j}}{g \cdot \ucov{x_j}}$. But then
    \[ p|_{\ballu\delta y} \colon \ballu\delta y 
       \longrightarrow p\bigl( \ballu\delta y\bigr)\]
    is also an isometric diffeomorphism. 
    Moreover, \[\ball{\delta} x \subset \ball{\varepsilon_{x_j}}{x_j} =
    p \bigl( \ballu{\varepsilon_{x_j}}{\ucov{x_j}}\bigr),\] 
    since $x \in
    \ball{\varepsilon_{x_j}/2}{x_j}$. Therefore
    $p\bigl(\ballu\delta
    y\bigr) = \ball\delta{p(y)}$. 
  \end{bew}

  How small does $\varepsilon$ have to be? 
  We take $\varepsilon \in \bigl(0, \min(C/4, \delta/3)
  \bigr)$ so small that $\vtilepsi \subset U^{\{e_0\}}$, where $C$ is the
  constant of Lemma~\ref{covdist} and $\delta$ is constructed in
  Lemma~\ref{werder}. 

  Since $\varepsilon < \delta$, we conclude that $P(\vtilepsi) \subset
  \vepsi$: 
  Let $\tau \in \vtilepsi$ and $x \in \ssim k$. Since $\rho(x) \in
  p^{-1}(K)$ and $\varepsilon < \delta$, it follows that
  \[ d_M\bigl( p \circ \tau(x), p \circ \rho(x)
        \bigr)
   = d_{\ucov M}\bigl( \tau(x), \rho(x) 
                \bigr)
   < \varepsilon. \]
  Hence $P(\tau) = p \circ \tau \in \vepsi$.

  Conversely, let $\tau \in \vepsi$. Since $\ssim k$ is simply
  connected, there is a lift $\ucov \tau$ of~$\tau$. Moreover, we may
  choose $\ucov \tau(e_0)$ in such a way that
  \[ d_{\ucov M}\bigl( \ucov \tau(e_0), \rho(e_0)\bigr) <
     \varepsilon\]
  because $\rho(e_0) \in p^{-1}(K)$ and $d_M\bigl(\tau(e_0), p \circ
  \rho(e_0) \bigr) < \varepsilon < \delta$.
  In order to show that $\ucov \tau \in \vtilepsi$ we have to check that the set
  \[ D:= \bigl\{ x \in \ssim k
         \bigm|  d_{\ucov M}\bigl( \ucov \tau (x), \rho(x) \bigr) < \varepsilon
         \bigr\}\]
  equals $\ssim k$.

  By construction, $e_0 \in D$. In particular, $D$ is not empty. Since
  $\ssim k$ is connected, it suffices to prove that $D$ is open and
  closed:
  \begin{itemize}
    \item \emph{The set $D$ is open.} Let $x \in D$. Since $\ucov \tau$ and
          $\rho$ are continuous, there is an open neighbourhood~$W$
          of~$x$ such that
          \[ \fa{y \in W} d_{\ucov M}\bigl( \ucov \tau(y), 
                                            \ucov \tau(x) 
                                     \bigr) < \varepsilon
                          \quad\text{and}\quad
                          d_{\ucov M}\bigl( \rho(y),
                                            \rho(x)
                                     \bigr) < \varepsilon. \]
          Hence
          \begin{align*}
             d_{\ucov M}\bigl( \ucov \tau(y)
                             , \rho(y) \bigr)
           & \leq d_{\ucov M}\bigl( \ucov \tau(y), \ucov \tau(x)\bigr)
                + d_{\ucov M}\bigl( \ucov \tau(x), \rho(x)\bigr)
                + d_{\ucov M}\bigl( \rho(x)      , \rho(y)\bigr) \\
           & < \varepsilon + \varepsilon +\varepsilon \leq \delta
          \end{align*}
          for all $y \in W$.
          Now the construction of~$\delta$ yields
          \[ d_{\ucov M}\bigl( \ucov \tau(y), \rho(y) \bigr)
           = d_M\bigl( \tau(y), p\circ \rho(y)\bigr) < \varepsilon.\]
          Thus $W \subset D$, implying that $D$ is open.

    \item \emph{The set $D$ is closed.} Suppose $x \in \ssim k \setminus D$,
          i.e., $d_{\ucov M}(\ucov \tau(x), \rho(x)) \geq
          \varepsilon$. The condition $\varepsilon < \delta$ assures
          that $p$ is isometric on the $\varepsilon$-ball in~$\ucov M$
          around~$\rho(x)$ and that $p\bigl(
          \ballu\varepsilon{\rho(x)}\bigr) = \ball \varepsilon{p \circ
          \rho(x)}$. In particular, the
          ball~$\ballu\varepsilon{\rho(x)}$ contains a
          point~$y$ such that $p(y)=\tau(x)$ because $\tau \in V_\varepsilon$. Now 
          Lemma~\ref{covdist} yields
          \begin{align*}
            d_{\ucov M}\bigl( \ucov \tau(x) 
                            , \rho(x)
                       \bigr) 
          & \geq d_{\ucov M}\bigl( \ucov \tau(x) 
                            , y
                            \bigr)
               -d_{\ucov M}\bigl( y 
                            , \rho(x)
                            \bigr) \\
          & \geq C - \varepsilon.
          \end{align*}
          By continuity, the set $W \subset \ssim
          k$ of all~$z$ satisfying $d_{\ucov M}(\ucov \tau(z),
          \ucov\tau(x)) < \varepsilon$ and $d_{\ucov M}(\rho(z), \rho(x)) <
          \varepsilon$ is an open neighbourhood of~$x$. 
          Therefore, we obtain for all $z \in W$ 
          \begin{align*}
            d_{\ucov M}\bigl( \ucov \tau(z), \rho(z)\bigr)
          & \geq d_{\ucov M}\bigl( \ucov \tau(x), \rho(x)\bigr)
               - d_{\ucov M}\bigl( \ucov \tau(z), \ucov\tau(x)\bigr)
               - d_{\ucov M}\bigl( \rho(x), \rho(z)\bigr) \\
          & \geq C - \varepsilon - \varepsilon -\varepsilon 
            \geq \varepsilon.
          \end{align*}
          Hence $W \subset \ssim k \setminus D$, which shows that $D$
          is closed.
  \end{itemize}
  This proves the first part of Theorem~\ref{borelsection}.

  2. As for the second part note that we can cover $\map[norm]{\ssim
  k}{\ucov M}$ with countably many open sets $(V_n)_{n\in\N}$ on 
  which $P$ is a homeomorphism (e.g., one could take a countable 
  covering of~$\ucov M$ by small sets~$U$ and consider sets of the form 
  $U^{\{e_0\}}$). In particular, the sets $(P(V_n))_{n \in \N}$ are
  open in~$\map[norm]{\ssim k}M$. Via
  $W_0 := P(V_0)$ and
  \[ \fa{n \in\N} W_{n+1} := P(V_{n+1}) \setminus 
                             \bigcup_{%
                             \makebox[0pt]%
                             {$\scriptstyle{j \in \{0, \dots, n\}}$}} W_j
  \]
  we get a countable family $(W_n)_{n\in\N}$ of mutually disjoint
  Borel subsets in 
  $\map[norm]{\ssim k}{M}$ such that $P^{-1}|_{W_n}$ is well-defined
  and continuous for each~$n \in \N$. 
  Moreover, $\map[norm]{\ssim k}M$ is covered by the $(W_n)_{n \in \N}$
  because $P$ is surjective. 
  Putting all these maps together
  yields the desired Borel section of~$P$.
\end{bew}


\subsection{Bowen's argument}\label{bowen}

In the following, flaws in Bowen's preprint ``An Isometry Between
Measure Homology and Singular Homology'' \cite{bowen} are outlined,
showing that his approach does not work. In this section, we will use
the notation of his preprint. 

Bowen's idea is to first improve measure cycles (without changing the
represented class in homology) and then explicitly construct a
singular chain representing the same class. Both steps are designed
in such a manner that the norm is not increased. 

For the first step a map~$R \colon \mhch * M \longrightarrow \mhch *
M$ is constructed, mapping measure chains to these improved
chains. Bowen claims that this map~$R$ is chain homotopic to the
identity. However, it is easy to see that $R$ is not even a chain map
in general. 

Furthermore, the inductive construction of simplices for the second
step is not possible: e.g., let $\sigma$ be the 2-simplex wrapping
around~$S^2$ and mapping the boundary~$\bou \ssim 2$ to the point~$x_0
\in S^2$. Let $\tau$ be the 2-simplex which maps everything to~$x_0$
(in particular, $\sigma|_{\bou \ssim 2} = \tau|_{\bou \ssim 2}$). Then
$\sigma \simeq \tau$ relative to the vertices of~$\ssim 2$, but not 
$\sigma \simeq \tau$ relative to the boundary~$\bou \ssim 2$. 

This example also shows that, in general, $[j(c)] \neq [\mu]$: Let
$\mu := j(\sigma - \tau)$. This is a cycle and $\mu$
generates~$\hsingc 2 {S^2}\R \cong \R$. However, $\sigma \in
H(\tau)$. Therefore,
\[ j(c) = \mu\bigl( H(\sigma)\bigr) \cdot j(\tau) = (1 -1 ) \cdot j(\tau) 
        = 0. \]
In particular, $[j(c)] \neq [\mu]$. 



% --------------------------------------------------------
\section{Integrating measure homology chains}\label{integration}%
  \tindex{integration!of smooth singular simplices}%
  \tindex{smooth singular simplex!integration of}%
  \tindex{singular simplex!integration of}%
  \tindex{integration!of measure chains}%
  \tindex{measure chain!integration of}%

As in (smooth) singular homology theory (Lemma~\ref{svvol}), we would
like to introduce an integration process for measure chains to get
information about the measure homology classes represented by certain
cycles.

\begin{defi}\label{defint}%
  \dindex{integration!of smooth singular simplices}%
  \dindex{smooth singular simplex!integration of}%
  \dindex{singular simplex!integration of}
  Let $M$ be a smooth manifold without boundary.
  Let $k \in \N$, $\sigma \in \maps[norm]{\ssim k}{M}$  
  and $\omega$ be a smooth $k$-form  
  on~$M$. Then there is an extension~$(f,V)$ of~$\sigma$ and we define
  \nindex{$f}{$\int_\sigma \omega$}{integration of the form $\omega$ over
  the smooth singular simplex~$\sigma$}%
  \[ \int_\sigma \omega := \int_{\ssim k} f^* \omega,\]
  where $f^* \omega$ denotes the pulled back $k$-form on the 
  $k$-manifold~$V$.
\end{defi}

Integrating forms over smooth simplices is indeed well-defined
(i.e., independent of the chosen extension) since 
the differentials of all extensions coincide on~$T\Delta^k$ 
(Lemma~\ref{defdiff}). 
Hence the pulled back forms coincide over~$\ssim k$, implying that 
the integral is well-defined. 

\begin{bem}
If $M$ is an oriented Riemannian manifold without
boundary of dimension~$n$, 
we get for all $\sigma \in \maps[norm]{\ssim{n+1}}{M}$ via 
the generalised Stokes Theorem \cite[Theorem~16.10]{leesmooth} 
\[ \int_{\bou \sigma} \vol_M = \int_\sigma d\vol_M \]
(where $\vol_M$ is the volume form on~$M$). 
Since $d\vol_M$ is an $(n+1)$-form of the $n$-dimensional
manifold~$M$, it must be zero. Hence
\[ \int_{\bou \sigma} \vol_M = 0.\]
Thus the linear extension $ \smsingc n M \R \longrightarrow \R $ of
integrating the volume form over a smooth singular simplex induces a homomorphism
\begin{align*}
  \hsmsingc{n}{M}{\R} & \longrightarrow \R \\
                [c]   & \longmapsto     \int_c \vol_M
\end{align*}
on homology.
\end{bem}


The following lemma reveals the reason for using the \cone-topology 
on the mapping space~$\maps[norm]{\ssim k}{M}$ instead of the coarser 
compact open topology. The map described in the Lemma would 
\emph{not} be continuous when working with the compact open topology 
(since the pulled back form considerably depends on the
differential). 

\begin{lem}\label{simint}%
  Let $M$ be a smooth manifold without boundary, $k \in \N$,  
  and let $\omega$ be some smooth $k$-form on $M$. 
  Then the map
  \begin{align*}
    \maps[big]{\ssim k}{M} & \longrightarrow \R \\
                 \sigma    & \longmapsto     \int_\sigma \omega
  \end{align*}
  is continuous.
\end{lem}
\begin{bew}
  This is an immediate consequence of the definition of the
  $\cone$-topology and the compatibility of integration with certain
  limits \cite[Lemma~3 in~\S11.5]{ratcliffe}.
\end{bew}

\begin{koro}%
  Let $M$ be a smooth manifold without boundary, $k\in\N$, and let $\omega$ be a 
  smooth $k$-form on~$M$. For each $\mu \in \mhch k M$ the map
  \begin{align*}
    \Omega \colon 
    \maps[big]{\ssim k} M & \longrightarrow \R \\
                 \sigma   & \longmapsto     \int_\sigma \omega
  \end{align*}
  is $\mu$-integrable. 
\end{koro}
\begin{bew}
  Let $D$ be a compact determination set of~$\mu$. By the previous 
  lemma, $\Omega$ is continuous. In particular, $\Omega$ is Borel
  measurable and the restriction~$\Omega|_{D}$ is bounded.

  The mapping space $\map[norm]{\ssim k}{M}$ is Hausdorff since $M$ is 
  Hausdorff. 
  Thus $\maps[norm]{\ssim k}M$ is Hausdorff by Lemma~\ref{simfiner}. In 
  particular, the compact set~$D$ is closed and hence Borel. 
  This implies that
  \[ \overline\Omega := \chi_D \cdot \Omega\] 
  is a bounded measurable function (where $\chi_D$ stands for 
  the characteristic function of the Borel set~$D$).
  Moreover,  $\maps[norm]{\ssim k} M \setminus D$ must be 
  a $\mu$-null set and hence also a null set with respect 
  to $\mu^+$~and~$\mu^-$.

  Since $\mu$ has finite total variation, Lemma~\ref{tov} implies
  \begin{alignat*}{2} 
    \mu^+(D) & = \phantom{-} \mu(D \cap P) && < \infty, \\
    \mu^-(D) & =           - \mu(D \cap N) && < \infty,
  \end{alignat*}
  where $\maps[norm]{\ssim k}M = P \cup N$ is a Hahn decomposition 
  for~$\mu$. Therefore the characteristic function~$\chi_D$ is 
  $\mu^+$-integrable and $\mu^-$-integrable. 
  Since $\Omega$ is almost everywhere (with respect to both $\mu^+$ and $\mu^-$)
  dominated by 
  the integrable function $\chi_D \cdot \max_{x \in D} \Omega(x)$, 
  it is itself integrable with respect to~$\mu^+$ and~$\mu^-$.  
\end{bew}

Hence we can integrate forms over measure chains:

\begin{defi}%
 \dindex{integration!of measure chains}%
 \dindex{measure chain!integration of}
 Let $M$ be a smooth manifold without boundary, $k \in \N$, 
 and let $\omega$ be some smooth $k$-form on~$M$. For each 
 measure chain $\mu \in \mhch k M$ we define
 \nindex{$e}{$\int_\mu \omega$}{integration of the form~$\omega$ over the
 measure chain~$\mu$}% 
 \[ \int_\mu \omega
     := \int_{\maps[norm]{\ssim k} M} 
        \int_\sigma \omega \; d \mu(\sigma). \qedhere\]
\end{defi}

In fact, this integration is a generalisation of integration over smooth 
simplices in the following sense:

\begin{lem}\label{intcom}
  Let $M$ be a smooth manifold without boundary, $k \in \N$, and let 
  $\omega$ be some smooth $k$-form on~$M$. For each smooth chain
  $c \in \smsingc k M \R$ 
  \[ \int_{i_M(c)} \omega  = \int_c \omega \]
  holds, $i_M \colon \smsingc * M \R \hookrightarrow \mhch * M$
  being the inclusion described in Lemma~\ref{inclsmmh}.
\end{lem}
\begin{bew}
  Since integration is linear, we only need to consider smooth singular
  simplices~$\sigma \in \smsingc k M \R$. By definition of~$i_M$, 
  \begin{align*}
        \int_{i_M(\sigma)} \omega 
    & = \int_{\maps[norm]{\ssim k} M} \int_\tau \omega 
             \;d\bigl(i_M(\sigma)\bigr)(\tau) \\
    & = \int_\sigma \omega \cdot 1,
  \end{align*} 
  which proves the lemma.
\end{bew}

The above compatibility statement and Lemma~\ref{svvol} can be
combined to yield the following description of the top measure
homology group: 

\begin{satz}%
\label{mhdr}%
\tindex{integration!measure homology and}%
\tindex{measure homology!integration and}
  Let $M$ be an oriented closed connected Riemannian manifold of dimension~$n$, and
  let $\mu \in \mhch n M$ be a measure cycle. Then the measure homology 
  class~$[\mu]$ can be expressed as follows:
  \[ [\mu] = \frac{\int_\mu \vol_M}{\vol(M)}   \cdot \mhfcl{M} \in \hmh n M.\]
\end{satz}
\begin{bew}
 The transformation formula shows \cite[Lemma~4 in~\S11.5]{ratcliffe}
  \begin{align*} 
     \fa{\nu \in \mhch{n+1}{M}} 
        \int_{\bou \nu} \vol_M  & = \int_{\nu} d\vol_M = \int_{\nu} 0 = 0. 
  \end{align*} 
  Thus integration of the volume form induces a homomorphism 
  \begin{align*}
     \hmh n M & \longrightarrow \R \\
        [\nu] & \longmapsto     \int_\nu \vol_M.
  \end{align*} 
  Denoting the canonical inclusion $\smsingc n M \R \hookrightarrow 
  \mhch n M$ by $i_M$, we obtain the commutative diagram
  (compare Lemma~\ref{intcom})
  \begin{equation*}
    \xymatrix@=3em{%
      {\hsmsingc n M \R} \ar[r]^-{\int_{\cdot}\vol_M}  
                         \ar[d]_-{H_n(i_M)}               & {\R.} \\
      {\hmh n M}         \ar[ru]_-{\int_{\cdot}\vol_M}
    }
  \end{equation*}
  The vertical arrow is an isomorphism by Theorem~\ref{smshmh}
  mapping the fundamental class~$\smfcl M$ to the measure 
  homology fundamental class~$\mhfcl M$.
  
  According to Lemma~\ref{svvol}
  \[ \int_c \vol_M = \vol(M) \]
  holds, whenever $c \in \smsingc n M \R$ is a representative of~$\smfcl
  M$. Hence
  \[ \int_\mu \vol_M = \int_c \vol_M = \vol(M) \]
  for each measure cycle~$\mu$ representing the measure homology 
  fundamental class~$\mhfcl M$.
\end{bew}
















propprinc.tex
% main: diploma_main.tex
\section{Statement of the proportionality principle}\label{stpp}

The proportionality principle of simplicial volume reveals a
fascinating connection between the simplicial volume and the
Riemannian volume:

\begin{satz}[Proportionality Principle of Simplicial Volume]%
    \label{propprinc}%
    \tindex{proportionality principle!of simplicial volume}%
    \tindex{simplicial volume!proportionality principle}
    \svproptmp
\end{satz}

When speaking of a \defin{universal Riemannian covering map}\dindex{universal
Riemannian covering}, we mean a universal
covering map which is an orientation preserving local isometry.

Without loss of generality, we may assume that the universal
Riemannian covering spaces of $M$~and~$N$ are \emph{orientation
preservingly} isometrically isomorphic. Otherwise we just reverse the
orientation of $M$.

Thus it is clear that we even may assume that $M$ and $N$ 
have the \emph{same} universal 
Riemannian covering since the composition of an orientation preserving 
isometry and a universal Riemannian covering map again is a universal 
Riemannian covering map.

For convenience, we introduce the following notation, which will be used in 
later sections.

\begin{setup}\label{ppsetup}
  Let $M$ and $N$ be two oriented closed connected Riemannian manifolds 
  with common universal Riemannian covering $U$. The covering maps 
  $U \longrightarrow M$ and $U \longrightarrow N$ will be denoted by 
  $p_M$ and $p_N$\nindex{pM}{$p_M$}{}\nindex{pN}{$p_N$}{} respectively. 
  Furthermore, we write $n$ for the dimension 
  of~$M$ and~$N$.
\end{setup}

In order to prove the proportionality principle, we are looking for a
chain map $\csingc{*}{N}{\R} \longrightarrow \csingc{*}{M}{\R}$
inducing multiplication by~$\vol(N)/\vol(M)$ on the top homology group
without increasing the seminorm on homology. Then the inequality
\[ \frac{\vol(N)}{\vol(M)} \cdot \sv M \leq \sv N \]
would easily follow. Swapping the r\^oles of $M$ and $N$ would give 
the reverse inequality and hence prove the proportionality principle. 

Integration (for smooth simplices) shows that the naive approach
\[\sigma \longmapsto p_M \circ \ucov{\sigma}\]
($\ucov{\sigma}$ being some $p_N$-lift of~$\sigma$) would yield
the factor $\vol(N)/\vol(M)$ if it were a chain map (which it is
not).

\iindex{smearing}%
So we may try to apply some averaging operation (``smearing'') 
to eliminate this choice of 
lift to get a chain map. But since $\pi_1(N)$ is not finite in general, 
there is no direct way to do this. Using measure homology chains instead 
of singular chains, we can take advantage of the Haar measure on the locally 
compact group $\isomp(U)$ of orientation preserving isometries on~$U$, 
which contains~$\pi_1(N).$ However, to satisfy the finiteness 
condition for measure chains, the compact quotient $\pi_1(M)
\setminus \isomp(U)$ is more appropriate. To guarantee the norm
condition, the constructed measure on the quotient~$\pi_1(M)\setminus \isomp(U)$ is
scaled to a probability measure. 

The resulting chain map 
\[\smear_{N,M} \colon \smsingc * N \R \longrightarrow \mhch * M \] 
indeed does not increase the seminorm and 
induces the desired factor on the level of homology -- as can be seen
by integration. 

The smearing technique was invented by Thurston
\cite[page~6.8]{thurston}. Thurston used it primarily to prove
Theorem~\ref{hypp}, which he calls ``Gromov's Theorem'' (and which is
by Gromov described as ``Thurston's Theorem'').  Thurston also
sketches a proof of the proportionality principle
based on smearing \cite[page~6.9]{thurston}. However, he uses the
isometric isomorphism of Theorem~\ref{isomisom}, but gives no evidence
for it (in fact, a paper is cited which never appeared in the
suggested form).  A more detailed discussion of smearing is given in
Ratcliffe's book
\cite[\S11.5]{ratcliffe}. 

The proportionality constant in the proportionality principle is only
known in very few special cases such as hyperbolic manifolds
(Theorem~\ref{hypp}) or flat manifolds (Corollary~\ref{flat}).
\newpage

%---------------------------------------------------------
% --------------------------------------------------------
\section{Isometry groups}\label{isometrygroups}\tindex{isometry group}

As indicated in Section~\ref{stpp}, the principal underlying tool 
for smearing is the measure on a 
quotient of the group of orientation preserving isometries on the universal 
covering manifold induced by the Haar measure on this isometry
group. So we first examine this isometry group. 

Whenever we refer to a Riemannian manifold as a metric space, it is understood 
that the metric is the one induced by the Riemannian metric.

\begin{bem}\tindex{isometry}
  Let $(U,g)$ be a Riemannian manifold without boundary. A map~$f \colon 
  U \longrightarrow U$ is a Riemannian isometry (i.e., $f$ is smooth and 
  the differential $Tf$ transforms the Riemannian metric~$g$ 
  into $g$) if and only if it is a 
  metric isometry \cite[Theorem~I~11.1]{helgason}.
\end{bem}

\begin{defi}\label{igdef}
  Let $U$ be a Riemannian manifold (without boundary, but not necessarily 
  compact). 
  \begin{itemize}
    \item The \dindex{isometry group}\dindex{topology!on isomco@on $\isomco(U)$}%
          \defin{group of (metric) isometries on}~$U$, 
          written as~$\isomco(U)$\nindex{Isomco}{$\isomco(U)$}{isometry group
          with the compact open topology},
          is the set of all metric isometries $U \longrightarrow U$ with the 
          multiplication given by composition, endowed with the 
          subspace topology induced by the compact open topology 
          on~$\map{U}{U}$.
    \item If $U$ is oriented, the 
          \dindex{topology!on $\isomp(U)$}%
          \defin{group of orientation preserving isometries on} $U$, 
          denoted by~ $\isomp(U)$\nindex{Isomp}{$\isomp(U)$}{group of
          orientation preserving isometries, endowed with the \cone-topology}, 
          is the mapping space 
	  consisting of all orientation preserving isometries 
          $U \longrightarrow U$
          with the multiplication given by composition. The topology 
          on~$\isomp(U)$ is the subspace topology induced by the \cone-topology
          on~$\maps{U}{U}$. \qedhere
  \end{itemize}
\end{defi}

From the topologist's point of view it is of course much more comfortable 
to work with the group $\isomco(U)$. However, we are forced to use the 
finer \cone-topology on the space of smooth singular simplices 
to make the integration process in Lemma~\ref{simint} 
continuous.
Therefore -- as can be seen in Lemma~\ref{intcont} -- we also have to 
take the \cone-topology on the isometry group.
We will see a posteriori (Theorem~\ref{coiscone}) that the compact 
open topology and the
\cone-topology on the isometry groups coincide. But there does not
seem to exist a short way to prove this fact. 

Before working out basic properties of $\isomp(U)$, we collect some facts about
the coarser group $\isomco(U)$. Lemma~\ref{tisom} will be the
key for applying these results to~$\isomp(U)$.

\begin{satz}\label{igco}\tindex{isometry group}
  Let $U$ be a Riemannian manifold.
  \begin{enumerate}
    \item The group $\isomco(U)$ is indeed a \emph{topological} group with
          respect to the given topology. 
    \item The set $\isomco(U)$ is closed in $\map{U}{U}$. 
    \item The group $\isomco(U)$ is Hausdorff, locally compact, and 
          satisfies the second countability axiom.
    \item If $G$ is a group acting properly and cocompactly on~$U$ by 
          isometries, then $G$ is a discrete and cocompact subgroup 
          of~$\isomco(U)$.
  \end{enumerate}
\end{satz}
\begin{bew}
  1. A proof is given by, e.g., Helgason \cite[Theorem~IV~2.5]{helgason}.
  
  2. If $f \in \map U U \setminus \isomco(U)$, there are $x$, $y \in U$
     such that 
     \[ d\bigl( f(x), f(y) \bigr) \neq d(x,y),\]
     where $d$ denotes the metric on $U$ determined by the Riemannian
     metric. In particular, $x \neq y$. Since $d \colon U \times U
     \longrightarrow \R_{\geq 0}$ is continuous, there are open
     neighbourhoods $V$ and $W$ of $f(x)$ and $f(y)$ respectively  
     such that
     \[ \fa{v \in V} \fa{w \in W} d(v,w) \neq d(x,y).\]
     Furthermore, $\{x\}$~and~$\{y\}$ are obviously compact. Hence
     $V^{\{x\}} \cap W^{\{y\}}$ is an open neighbourhood of~$f$ in $\map U
     U$ satisfying
     \[ \bigl(V^{\{x\}} \cap W^{\{y\}}\bigr) 
      \cap \isomco(U) = \emptyset,\]
     by construction of $V$ and $W$.

  Parts 3.\ and 4.\ follow since Riemannian manifolds are second
  countable proper metric spaces \cite[Theorem~2.35]{roman}.  
  The cited proof is based on the Arzela-Ascoli Theorem, and for this
  to apply, it is crucial that $\isomco(U)$ is closed in $\map{U}{U}.$
\end{bew}

Since the actual object of interest is the group $\isomp(U)$, we would like to 
have a similar theorem for $\isomp(U)$. 
This will be accomplished by equipping the tangential space of~$U$  
with a Riemannian structure in a canonical way:

\begin{bem}\label{tangmetric}
  Let $U$ be a Riemannian manifold. On the tangent bundle $TU$, a Riemannian 
  metric can be defined as follows \cite[Chapter~3, Exercise~2]{docarmo}: 
  
  Let $(p,q) \in TU$ and $v$, $w \in T_{(p,q)}(TU)$. Choose smooth curves
  \begin{alignat*}{2}
    \alpha_v & =(p_v,q_v) && \colon [\,-1, 1\,] \longrightarrow TU, \\
    \alpha_w & =(p_w,q_w) && \colon [\,-1, 1\,] \longrightarrow TU
  \end{alignat*} 
  representing $v$ and $w$ respectively, i.e.,
  $p_v(0) = p = p_w(0)$, $q_v(0) = q = q_w(0)$ and
  $v = \alpha_v'(0)$, $ w = \alpha_w'(0)$. Then the inner product of 
  $v$~and~$w$ is given by
  \begin{align*}
    \langle v,w \rangle_{(p,q)} := \bigl\langle T_{(p,q)}\pi(v)  
                                              , T_{(p,q)}\pi(w)
                                   \bigr\rangle_p
                                 + \langle D_0 q_v
                                         , D_0 q_w
                                   \rangle_p
  \end{align*}
  where $\pi \colon TU \longrightarrow U$ is the bundle projection,  
  $D_0 q_v$ denotes the covariant derivative of the vector field~$q_v$ 
  along~$p_v$ (with respect to the Riemannian connection, of course),
  and analogously for~$D_0 q_w$.
\end{bem}

\begin{lem}\label{tisom}
  Let $U$ be an oriented Riemannian manifold and $f \in \isomp(U)$. Then 
  \[ Tf \in \isomco(TU).\]
\end{lem}
\begin{bew}
  The construction of the smooth structure on the tangent bundle $TU$
  shows that $Tf \colon TU \longrightarrow TU$ is smooth. Therefore it
  remains to show that $Tf$ preserves the Riemannian metric on~$TU$:

  Suppose $(p,q) \in TU$ and $v$, $w \in T_{(p,q)}(TU)$. Choose smooth curves
  \begin{align*}
    (p_v,q_v) & \colon [\,-1, 1\,] \longrightarrow TU, \\
    (p_w,q_w) & \colon [\,-1, 1\,] \longrightarrow TU
  \end{align*} 
  representing $v$ and $w$ respectively. Using the chain rule and the
  fact that $\pi \circ Tf = f \circ \pi$, we conclude
  \[ T_{(f(p),T_pf(q))} \pi\bigl(T_{(p,q)} Tf(v)\bigr) 
   = T_p f\bigl(T_{(p,q)} \pi (v)\bigr).\]
  Since $f$ is an isometry of the Riemannian manifold $U$, this yields
  \begin{align*}
  &  \bigl\langle T_{(f(p),T_pf(q))} \pi\bigl(T_{(p,q)} Tf(v)\bigr)
                , T_{(f(p),T_pf(q))} \pi\bigl(T_{(p,q)} Tf(w)\bigr)
     \bigr\rangle_{f(p)} \\
  &\qquad= \bigl\langle T_p f\bigl(T_{(p,q)} \pi (v)\bigr)
                , T_p f\bigl(T_{(p,q)} \pi (w)\bigr)
          \bigr\rangle_{f(p)} \\
  &\qquad= \langle T_{(p,q)}\pi(v)  
                 , T_{(p,q)}\pi(w)
          \rangle_p.
  \end{align*}
  Considering the second summand in the definition of~$\langle v,w \rangle_{(p,q)}$, 
  we observe that (by naturality of the 
  Riemannian connection under isometries \cite[Proposition~(5.6)]{lee})
  \[ D_0(Tf \circ q_v) = Tf(D_0 q_v) \]
  which implies (since $f$ is an isometry)
  \begin{align*}
      \bigl\langle D_0(Tf \circ q_v)
                 , D_0(Tf \circ q_w)
      \bigr\rangle_{f(p)}
  & = \bigl\langle Tf(D_0 q_v)
                 , Tf(D_0 q_w)
      \bigr\rangle_{f(p)} \\
  & = \langle D_0 q_v
            , D_0 q_w
      \rangle_p.
  \end{align*}
  Thus the Riemannian metric on $TU$ is preserved by the map $Tf$, i.e., $Tf$ 
  is an isometry of $TU$.
\end{bew}

In view of Lemma~\ref{tisom} we can use the results and methods of 
Theorem~\ref{igco} to derive a corresponding theorem for the group~$\isomp(U)$:

\begin{satz}\label{ig}\tindex{isometry group}
  Let $U$ be an oriented Riemannian manifold.
  \begin{enumerate}
    \item The group $\isomp(U)$ is indeed a \emph{topological} group with 
          respect to the topology from Definition~\ref{igdef}.
    \item The group $\isomp(U)$ is Hausdorff and satisfies the second 
          countability axiom. 
    \item The set $T\bigl(\isomp(U)\bigr)$ is closed 
          in~$\maps{TU}{TU}$. 
    \item Moreover, the group $\isomp(U)$ is locally compact.
    \item If $G$ is a group acting properly and cocompactly on $U$ by 
          isometries, then $G$ is a discrete and cocompact subgroup 
          of~$\isomp(U)$.
  \end{enumerate}
\end{satz}
\begin{bew}
  Lemma~\ref{tisom} shows that $T\bigl(\isomp(U)\bigr) \subset \isomco(TU)$.
  Therefore 
     \[ T \colon \isomp(U) \longrightarrow 
                 T\bigl(\isomp(U)\bigr) \subset \isomco(TU)\]
  is (by definition of the topology on $\isomp(U)$) a homeomorphism.   

  1. Since multiplication and inverting elements on $\isomco(TU)$ are
     continuous according to Theorem~\ref{igco}, this is also true for the restrictions
     to the subset~$T\bigl(\isomp(U)\bigr)$. Thus $\isomp(U)$ is a topological
     group.

  2. Since $\isomco(TU)$ is Hausdorff by Theorem~\ref{igco}, the same is 
     true for the subset~$T\bigl(\isomp(U)\bigr)$. Therefore
     $\isomp(U)$ is also Hausdorff. By the same argument, $\isomp(U)$  
     is second countable. 


  3. Using the fact that $\isomco(TU)$ is Hausdorff and second countable, 
     we can check the closedness of~$T\bigl(\isomp(U)\bigr)$ via 
     sequences:

     Let $(g_n)_{n \in \N} \subset T\bigl(\isomp(U)\bigr)$ be a sequence 
     converging to $g \in \isomco(TU)$. Hence we can write 
     \[ \fa{n \in \N} g_n = Tf_n\]  
     for an appropriate sequence~$(f_n)_{n \in \N} \subset \isomp(U)$. The 
     convergence of the sequence~$(Tf_n)_{n \in\N}$ to~$g$ 
     implies that $(f_n)_{n \in \N}$ converges to 
     \[ \overline g := \pi_U \circ g \circ i_U\] 
     in 
     the compact open topology (where $i_U \colon U \longrightarrow TU$ is 
     the inclusion of the zero-section and $\pi_U \colon TU \longrightarrow 
     U$ denotes the bundle projection). Since $i_U$~and~$\pi_U$ are 
     smooth, it follows that $\overline g$ is smooth. 
     Theorem~\ref{igco} implies thus that $\overline g$ is an isometry. 

     To convince ourselves that $T\overline g = g$, we use (special) charts 
     of~$TU$ to reduce the problem to a similar one in Euclidean space
     (compare Lemma~\ref{convtech}), where it can easily be solved.
     
     Let $(V_1, \varphi_1)$ be a relatively compact chart around~$x
     \in U$ (i.e., the closure of~$V_1$ is compact)  
     and let $(V_2, \varphi_2)$ be a chart around $\overline g(x) \in
     U$ such that $\overline g(V_1) \subset V_2$. 
     Since each of the $g_n = Tf_n$ preserves fibres, the same must be true 
     for their limit $g$. Then
     \[ g \bigl( \pi_U^{-1} (V_1)\bigr) 
        \subset \pi_U^{-1} (V_2). \]

     By definition of the manifold structure on~$TU$, the pairs 
     $\bigl( \pi_U^{-1}(V_1), T\varphi_1 \bigr)$ and
     $\bigl( \pi_U^{-1}(V_2), T\varphi_2 \bigr)$
     are charts of~$TU$ around each point in $\pi_U^{-1}(x)$ and 
     $\pi_U^{-1}(\overline g(x))$ respectively.

     Since $V_1$ is relatively compact and $V_2$ is open, $f_n (V_1) 
     \subset V_2$ holds for all large enough $n \in \N$. So, without loss 
     of generality, we may assume that this is true for all $n \in \N$. 
     This implies
     \[ Tf_n\bigl( \pi_U^{-1}(V_1)\bigr) 
        \subset \pi_U^{-1}(V_2)\]
     for all $n \in \N$ since $Tf_n$ is fibre-preserving. Thus we may define
     \begin{align*}
       G   & := T\varphi_2 \circ g    \circ (T\varphi_1)^{-1}, \\
       F_n & := T\varphi_2 \circ Tf_n \circ (T\varphi_1)^{-1}. 
     \end{align*}
     In particular, we get commutative diagrams of the form (with  
     $d := \dim(U)$)
     \begin{equation*}
       \xymatrix@=3em{%
          {\pi_U^{-1}(V_1)} \ar[d]_{T\varphi_1} \ar[r]^{g_n|=Tf_n|} 
        & {\pi_U^{-1}(V_2)} \ar[d]^{T\varphi_2}                   \\
        % --------------------------------------------------------
          {\varphi_1(V_1) \times \R^d}          \ar[r]^{F_n}
        & {\varphi_2(V_2) \times \R^d.}
       }
     \end{equation*}
     On the other hand, the differential $Tf_n$ is characterised by the 
     fact that the above diagram commutes when the map on the bottom is 
     substituted by
     \begin{align*}
       \varphi_1(V_1) \times \R^d & \longrightarrow \varphi_2(V_2) \times \R^d\\
                    (y,z)         & \longmapsto
                    \bigl( (\varphi_2 \circ f_n \circ \varphi_1^{-1})(y)
                         , J(\varphi_2 \circ f_n \circ \varphi_1^{-1})_y(z)
                    \bigr)
     \end{align*}
     (here $J$ stands for the Jacobian matrix).
     Hence $F_n$ must be this (smooth) map (the vertical arrows are 
     diffeomorphisms).
     Moreover, $(F_n)_{n \in \N}$ converges to~$G$ in the compact open 
     topology since $T\varphi_1$ and $T\varphi_2$ are homeomorphisms.

     Thus we can apply the following lemma
     (whose proof is deferred until the end of the proof of the theorem):

     \begin{lem}\label{convtech}
       Let $d \in \N$, let $V_1$, $V_2 \subset \R^d$ be two open sets, and 
       $(f_n)_{n \in\N} \subset \maps{V_1}{V_2}$. Assume that the maps 
       $(F_n)_{n \in \N} \subset \maps{V_1 \times \R^d}{V_2 \times \R^d}$, 
       defined by 
       \begin{align*}
          F_n \colon V_1 \times \R^d & \longrightarrow V_2 \times\R^d \\
                     (y,z)           & \longmapsto     
                                       \bigl( f_n(y)
                                            , (Jf_n)_y(z)\bigr)
       \end{align*}
       converge in the compact open topology to 
       a function~$G \colon V_1 \times \R^d \longrightarrow V_2 \times \R^d$. 
       Then $G_1 \colon V_1 \longrightarrow V_2$ is continuously differentiable 
       and 
       \[ \fa{(y,z) \in V_1 \times \R^d}
          (JG_1)_y (z) = G_2(y,z),\]
       where (denoting the obvious inclusions/projections by~$i_1$, $p_1$, 
       $p_2$)
       \begin{align*}
         G_1 & := p_1 \circ G \circ i_1,\\
         G_2 & := p_2 \circ G.
       \end{align*}
     \end{lem}
 
     Since the image of the zero-section is closed in~$TU$ and preserved by all 
     the $g_n = Tf_n$, the same must hold for $g$. Then a straightforward 
     computation shows  
     \[    \varphi_2 \circ \overline g \circ (\varphi_1)^{-1} 
         = p_1 \circ G \circ i_1 
         =: G_1.\]
     Therefore the lemma shows
     \[ \fa{(y,z)\in \varphi_1(V_1) \times \R^d}
        (J\overline g)_y (z) = p_2 \circ G (y,z) =: G_2(y,z),\]
     yielding the commutative diagram
     \begin{equation*}
       \xymatrix@=3em{%
          {\pi_U^{-1}(V_1)} \ar[d]_{T\varphi_1} \ar[r]^{g|} 
        & {\pi_U^{-1}(V_2)} \ar[d]^{T\varphi_2}                   \\
        % --------------------------------------------------------
          {\varphi_1(V_1) \times \R^d}          \ar[r]^{(G_1
                                                      , JG_1)}
        & {\varphi_2(V_2) \times \R^d.}
       }
     \end{equation*}
     On the other hand, we already know that $\overline g$ is smooth. So the above 
     diagram proves
     \[ g|_{\pi_U^{-1}(V_1)} =  T\overline g|_{\pi_U^{-1}(V_1)}.\]
 
     Moreover, the map $\overline g$ is orientation preserving: 
     A local diffeomorphism between 
     oriented smooth manifolds is orientation preserving if and only
     if its Jacobian 
     matrix with respect to any oriented smooth charts has positive 
     determinant \cite[Exercise 13.5]{leesmooth}. 
     Using the same argument as above, we get that the 
     Jacobians of the $(f_n)_{n \in \N}$ converge pointwise to  
     the Jacobian of $\overline g$ with respect to such charts. Since the 
     determinant is continuous and each $f_n$ preserves the orientation, 
     their limit~$\overline g$ must also be orientation preserving.

  4. To see that $\isomp(U)$ is locally compact, we apply
     Theorem~\ref{igco} to the group~$\isomco(TU)$. Thus for each map~$f \in
     \isomp(U)$ the differential~$Tf \in \isomco(TU)$ possesses a
     compact neighbourhood~$V$. Then $V \cap T\bigl( \isomp(U)\bigr)$
     is a compact neighbourhood of~$Tf$ in~$T\bigl(\isomp(U)\bigr)$ because  
     $T\bigl(\isomp(U)\bigr)$ is a closed subset of the Hausdorff
     space~$\isomco(TU)$. Therefore 
     \[ T^{-1}\bigl(V \cap T\bigl( \isomp(U)\bigr)\bigr)\]
     is a compact neighbourhood of $f$ in $\isomp(U)$.

  5. Due to Theorem~\ref{igco}, we know that $G$ is a discrete subgroup 
     of~$\isomco(U)$. Since the topology on $\isomp(U)$ is finer than the 
     compact open topology, it is clear that~$G \subset \isomp(U)$ is 
     discrete.
     
     We now proceed as in Sauer's proof of Theorem~\ref{igco}
     \cite[Theorem~2.35]{roman}:  
     Choose a point $x \in U$. Since $G$ acts cocompactly on $U$, we
     can find a compact set $K \subset U$ such that \cite[Lemma~VIII~3.19]{elstrodt}
     \[ G \cdot K = M.\] 
     Hence for every isometry $f \in \isomp(U)$ there exists an 
     element~$g_f \in G$ such that $g_f \circ f(x) \in K$. We consider 
     the set
     \[ K_G := \bigl\{    T(g_f \circ f) 
               \,\bigm|\, f \in \isomp(U)
               \bigr\}.
     \] 
     According to Lemma~\ref{tisom}, we know that all the elements of $K_G$ have 
     Lipschitz constant $1$. Moreover, 
     \begin{align*}
      K_G(0_x) & := \bigl\{    T(g_f \circ f) (0_x)
                \,\bigm|\, f \in \isomp(U)
                \bigr\} \\
      &\subset  \bigl\{    0_y 
                \,\bigm|\, y \in \pi^{-1}(K)
                \bigr\} \\
      & =       i(K),
     \end{align*}
     where $\pi \colon TU \longrightarrow U$ denotes the bundle 
     projection, $0_y$ (for $y\in U$) is the zero element of the 
     tangent space $T_yU$ and $i \colon U \longrightarrow TU$ is the
     zero section. Thus $K_G(0_x)$ is relatively compact. 
     
     Therefore, we can apply 
     the Arzela-Ascoli Theorem \cite[Lemma~2.13]{roman}, 
     which shows that $K_G$ is relatively compact in~$\map{TU}{TU}$.
     Since the image $T\bigl(\isomp(U)\bigr)$ is a closed subset of the Hausdorff 
     space~$\maps{TU}{TU}$, it follows that $K_G \cap T\bigl(\isomp(U)\bigr)$ 
     is relatively compact in~$T\bigl(\isomp(U)\bigr)$.
     Hence 
     \[ T|_{\isomp(U)}^{-1}(K_G) = \{    g_f \circ f 
                      \,|\, f \in \isomp(U) 
                      \}
     \] 
     must be relatively compact in~$\isomp(U)$.
     
     Denoting the canonical projection $\isomp(U) \longrightarrow G \setminus 
     \isomp(U)$ by $\pi_G$, we get by 
     construction that the restriction 
     \[ \pi_G|_{T|_{\isomp(U)}^{-1}(K_G)} \colon  T|_{\isomp(U)}^{-1}(K_G) \longrightarrow 
                                     G \setminus \isomp(U) \]
     is surjective, implying the compactness of the 
     quotient~$G\setminus \isomp(U)$.
\end{bew}

To complete the proof of the previous theorem, it remains show   
Lemma~\ref{convtech}:

\begin{bew}[Proof (of Lemma~\ref{convtech}).]
  Convergence in the compact open topology coincides with uniform convergence 
  on compact subsets (Remark~\ref{uniform}). 
  In particular, the~$f_n$ converge pointwise to~$G_1$
  for $n \rightarrow  \infty$.

  Since each $F_n$ preserves fibres and is linear in each fibre, the same 
  holds for their limit~$G$. If $K \subset V_1$ is compact, the product 
  $K \times S^{d-1}$ is a compact subset of~$V_1 \times \R^d$. Hence
  the $F_n$ converge  uniformly on~$K \times S^{d-1}$ to~$G$ for $n 
  \rightarrow \infty$. This implies that
  \[ Jf_n \longrightarrow G_2 \]
  uniformly on~$K$ for $n \rightarrow \infty$ (with respect to the operator 
  norm).

  A standard theorem from calculus \cite[Theorem A.70]{leesmooth} 
  now shows that $G_1$ is continuously differentiable and that 
  \[ \fa{(y,z) \in V_1 \times \R^d}
      (JG_1)_y(z) = G_2(y,z). \qedhere\]
\end{bew}

\begin{koro}\label{quotcomp}
  Let $M$ be an oriented Riemannian manifold without boundary 
  and $U$ its Riemannian 
  universal covering. 
  Then $\pi_1(M)$  can be considered a subgroup of~$\isomp(U)$ and the
  quotient $\pi_1(M) \setminus \isomp(U)$ is compact.
\end{koro}
\begin{bew}
  Covering theory shows that $\pi_1(M)$ acts on $U$ from the right by
  (mutually distinct) homeomorphisms. 
  Since the covering map $\pi_M \colon U \longrightarrow M$ is an 
  orientation preserving local isometry and 
  \begin{equation*}
    \xymatrix@=3em{%
      {U} \ar[r]^{\cdot\alpha} \ar[d]_{\pi_M}  & {U} \ar[d]_{\pi_M}\\
      {M} \ar[r]_{\id}                         & {M}
    }
  \end{equation*} 
  is commutative for each $\alpha \in \pi_1(M)$, it follows that 
  $\pi_1(M)$ acts on~$U$ by orientation preserving isometries. 
  Moreover, multiplication in~$\pi_1(M)$ corresponds to composition
  in~$\isomp(U)$ \cite[Corollary~V~7.5]{massey}. 
  Hence $\pi_1(M)$ is a subgroup of~$\isomp(U)$. 

  The quotient~$\pi_1(M) \setminus \isomp(U)$ is compact:
  By covering theory, $\pi_1(M)$ acts properly on~$U$  
  \cite[page~136]{massey} and $\pi_1(M) \setminus U \cong M$ 
  is compact \cite[Lemma~8.1 in Chapter~V]{massey}.
  Thus $\pi_1(M) \setminus \isomp(U)$ is compact by the last part of  
  Theorem~\ref{ig}.
\end{bew}

Using the results of Theorem~\ref{ig}, we will now be able to conclude that the compact
open and the \cone-topology on the isometry groups
coincide (but we will not need this fact in the sequel). Unfortunately, 
I was not able to figure out a proof of this fact which avoids the
nasty computations in the proof of Theorem~\ref{ig}.

\begin{defi} 
  For a Riemannian manifold $U$, we write $\isom(U)$% 
  \nindex{Isom}{$\isom(U)$}{group of isometries, endowed with the \cone-topology}%
  for the group of
  isometries\dindex{isometry group}\dindex{topology!on $\isom(U)$}  
  endowed with the subspace topology induced by the
  \cone-topology on~$\maps U U$. 
\end{defi}

\begin{satz}\label{coiscone}\tindex{isometry group}
  Let $U$ be a Riemannian manifold. Then the topologies on
  the groups $\isomco(U)$ and $\isom(U)$ are the same.
\end{satz}
\begin{bew}
  Exactly the same methods as in the proof of Theorem~\ref{ig} show
  that $\isom(U)$ is a topological group  which is Hausdorff, locally
  compact, and second countable. 

  We will now make use of the methods described in Kechris' book ``Classical Descriptive
  Set Theory'' \cite{kechris}. One of the central notions is
  that of a standard Borel space, which provides a convenient
  setting for measure theory. 

  As locally compact second countable Hausdorff groups, both
  $\isomco(U)$ and $\isom(U)$ are standard Borel groups
  \cite[Theorem~5.3]{kechris}.  Since the \cone-topology is finer than
  the compact open topology, the identity map $\id\colon\isom(U)
  \longrightarrow \isomco(U)$ is a bijective Borel map. Then $\id$
  already is a Borel isomorphism \cite[Theorem~14.12]{kechris}.  Hence
  $\isom(U)$ and $\isomco(U)$ coincide as Borel spaces.

  This suffices to conclude that the topologies on
  $\isom(U)$~and~$\isomco(U)$ are the same
  \cite[Proposition~12.25]{kechris}.
\end{bew}

% --------------------------------------------------------
\section{The Haar measure}\label{haarmeasure}

In the following, some basic properties of Haar measures are recalled. 
Haar measures are regular measures on locally compact groups which are 
compatible with the group structure. 
Using the results from the previous section, we will prove that there 
is a right invariant probability measure on the Borel $\sigma$-algebra
of the quotient~$\pi_1(M) \setminus \isomp(U)$. This measure will be
used in the averaging process mentioned in the introduction.

\begin{defi} 
  Let $G$ be a locally compact Hausdorff topological group, and let
  $\siga$ be its Borel~$\sigma$-algebra. 
  \begin{itemize} 
  \item A (positive) measure~$\mu$ on~$\siga$ is called
        \defin{regular}\dindex{regular}  
        if 
        \begin{align*}
          \mu(A) & = \sup\bigl\{ \mu(K)
                         \bigm|  \text{$K$ compact, 
                                       $K \subset A$}
                         \bigr\}, \\
          \mu(A) & = \inf\bigl\{ \mu(U)
                         \bigm|  \text{$U$ open, $A \subset U$}
                         \bigr\}
        \end{align*}
        holds for all $A \in \siga$, and if $\mu(K)$ is finite for all
        compact sets~$K \subset G$.         
  \item A \defin{Haar measure on}~$G$\dindex{Haar measure} is a positive  
        measure $h$ on~$\siga$ which is regular, 
        non-zero on any non-empty open set, and which is 
        \defin{left invariant}\dindex{left invariant}, 
        i.e., 
        \[ h(g \cdot A) = h(A) \]
        for all $g \in G$ and all $A \in \siga$.
  \item The group $G$ is called \defin{unimodular}\dindex{unimodular} 
        if each left
        invariant Haar measure on~$G$ also is right invariant. 
        \qedhere  
  \end{itemize}
\end{defi}

It is well-known that such a Haar measure exists for each locally compact 
Hausdorff 
group and is essentially unique (up to multiplication with a positive 
constant) \cite[Satz~VIII~3.12]{elstrodt}. 

For example, the Haar measure on the discrete group~$\Z$ is the counting 
measure, and the Haar measure on the additive group~$\R$ is the Lebesgue 
measure. Obviously, all Abelian groups are unimodular. Additionally, all 
compact groups are unimodular \cite[Satz~VIII~3.16]{elstrodt}, 
and -- as we will show -- the isometry group~$\isomp(U)$ of an oriented 
Riemannian manifold is unimodular. 

\begin{defi}
  Let $X$ be a topological space, let $\mu$ be a measure on the
  Borel~$\sigma$-algebra of~$X$, and suppose that the group~$G$
  acts via Borel isomorphisms  on~$X$ (from the left). A \defin{measure
  fundamental domain}\dindex{measure fundamental domain} 
  of this action is a Borel set $F \subset X$ such
  that 
  \[ \mu(X \setminus G \cdot F) = 0, \]
  and 
  \[ \fa{g \in G \setminus\{1\}} 
     \mu(g\cdot F \cap F) = 0. \qedhere\]
\end{defi}

\begin{satz}\label{measure}\tindex{isometry group!measure on}
  Presuppose the notation from Setup~\ref{ppsetup}. 
  There exists a (positive) right invariant measure~$h_M$ on the 
  Borel~$\sigma$-algebra of~$\pi_1(M) \setminus
  \isomp(U)$ satisfying 
  \nindex{hxM}{$h_M$}{}%
  \[ h_M\bigl( \pi_1(M) \setminus \isomp(U) \bigr) = 1.\]
\end{satz}
\begin{bew}
  Using properties of standard Borel spaces, it can be shown
  that each discrete cocompact subgroup of a
  locally compact second countable Hausdorff group~$G$ has
  \defin{finite covolume} with respect to the Haar measure (i.e.,
  there is a measure fundamental domain of finite measure for this
  subgroup) \cite[page~42]{roman}. 
  Hence $G$ must be unimodular \cite[Lemma~2.32]{roman}.

  Since $\isomp(U)$ is a locally compact second countable Hausdorff
  topological group (Theorem~\ref{ig}) and $\pi_1(M) \subset
  \isomp(U)$ is discrete and acts cocompactly on~$\isomp(U)$
  (Corollary~\ref{quotcomp}), we conclude that $\pi_1(M)$ has finite
  covolume in~$\isomp(U)$ and that $\isomp(U)$ is unimodular. 

  It follows that the Haar measure~$h$ on~$\isomp(U)$ descends 
  to a right invariant measure~$h'$ on the
  quotient $\pi_1(M) \setminus \isomp(U)$ 
  \cite[Korollar~3.25 in Kapitel VIII]{elstrodt}. 
  If $F$ is a finite measure
  fundamental domain of~$\pi_1(M)$ and $\pi_M \colon \isomp(U)
  \longrightarrow \pi_1(M) \setminus \isomp(U)$ is the projection,
  then one can show that 
  \[ h'(A) = h \bigl( \pi_M^{-1}(A) \cap F\bigr)\]
  holds for all Borel subsets $A \subset \pi_1(M) \setminus
  \isomp(U)$: 
 
  Indeed, for each $g \in \pi_1(M)$ the translate~$F\cdot g$ is also a
  measure fundamental domain of finite measure because $h$ is right
  invariant. Left invariance of~$h$ yields that
  $h\bigl(\pi_M^{-1}(\,\cdot\,) \cap F\bigr)$ is independent of the 
  chosen fundamental domain~$F$. Then it is easy to see that 
  $h \bigl(\pi_M^{-1}(\,\cdot\,) \cap F\bigr)$ is a right invariant
  measure on the quotient~$\pi_1(M) \setminus \isomp(U)$. 

  Since $\pi_1(M)$ has finite covolume in~$\isomp(U)$, it follows that 
  $h'\bigl( \pi_1(M) \setminus \isomp(U)\bigr)$ is finite. 
  Why is  $h'\bigl( \pi_1(M) \setminus \isomp(U)\bigr) \neq 0$? 
  Since $\pi_1(M)$ is countable and $F$ is a measure fundamental
  domain, we get (where the number of non-zero summands might be
  infinite) 
  \begin{align*}
    h\bigl( \isomp(U)\bigr)
      & = \sum_{g \in \pi_1(M)} h(g \cdot F) \\
      & = \sum_{g \in \pi_1(M)} h(F).
  \end{align*} 
  In addition, $h\bigl( \isomp(U)\bigr) \neq 0$ since Haar measures
  are by definition nontrivial. Thus
  \[ 0 \neq h(F) = h'\bigl( \pi_1(M) \setminus \isomp(U) \bigr).\]
  So 
  \[ h_M := \frac{h'}{ h'\bigl( \pi_1(M) \setminus \isomp(U)\bigr)}\]
  is a measure with the desired properties. 
\end{bew}


% --------------------------------------------------------
\section{Smearing}\label{smearing}\tindex{smearing}

Using the methods of the preceding sections, we are now able to 
introduce the main construction -- the ``smearing'' of smooth 
singular chains.

In this section we will always presuppose the notation from Setup~\ref{ppsetup}.
As a first step, we define for each smooth simplex~$\sigma$ on~$U$ a
measure $\smear_M(\sigma)$ which is supported on all $p_M \circ\tau$, 
where $\tau$ is an (orientation preserving) isometric translate 
of~$\sigma$ and which is uniformly distributed -- the simplex~$\sigma$
is ``smeared'' over the set $p_M\circ\bigl(\isomp(U)
\cdot \sigma\bigr)$.

Subsequently, we will pull back this construction to smooth singular simplices 
on~$N$. Using the results from Section~\ref{integration}, we will show that 
smearing induces indeed the desired factor $\vol(N)/ \vol(M)$ on the 
top homology. Moreover, it will be fundamental that smearing does not 
increase the norm.

\begin{lem}\label{intcont}
  Let $k\in \N$ and $\sigma \in \maps[norm]{\ssim k}{U}$. The map
  \nindex{Fsigma}{$F_\sigma$}{}%
  \begin{align*}
     F_\sigma \colon \pi_1(M)\setminus\isomp(U) 
   & \longrightarrow \maps[big]{\ssim k}{M} \\
     \bigl(\pi_1(M)\bigr) \cdot f 
   & \longmapsto p_M \circ f \circ \sigma               
  \end{align*}
  is well-defined and continuous.
\end{lem}

Since we take the \cone-topology on the right hand side, we are 
forced to consider the \cone-topology on the isometry group~$\isomp(U)$ 
to get a continuous map.

\begin{bew}
  For all $\alpha \in \pi_1(M)$ we get 
  \[ p_M \circ(\cdot \alpha) = p_M \]
  since $\pi_1(M)$ acts by deck transformations, implying that
  $F_\sigma$ is independent of the chosen representative.

  By Lemma~\ref{comp} and Remark~\ref{smmapssim} the map
  \begin{align*}
    \isomp(U) & \longrightarrow \maps[big]{\ssim k} M \\
            f & \longmapsto     p_M \circ f \circ \sigma
  \end{align*}
  is continuous. Thus the universal property of the quotient topology 
  yields the continuity of~$F_\sigma$.
\end{bew}


\begin{defi}
  For each $k \in \N$ and each  $\sigma \in
  \maps[norm]{\ssim k}U$, we define the \defin{smeared chain
  of}~$\sigma$\dindex{smearing}\dindex{chain!smeared} by
  \nindex{smear}{$\smear_M$}{first step of smearing\\}% 
  \[ \smear_M(\sigma) := h_M^{F_\sigma}, \]
  where $h_M$ is the measure from Theorem~\ref{measure}. 
  We extend $\smear_M$ linearly to the whole chain group~$\smsingc k U
  \R$.
\end{defi}

By Lemma~\ref{intcont}, the map $F_\sigma$ is, in particular, Borel measurable. 
Hence $\smear_M(\sigma)$ is a well-defined measure
on~$\maps[norm]{\ssim k}{M}$. 

\begin{lem}\label{smuch}
  \begin{enumerate}
    \item For each $k \in \N$ and each smooth chain $c \in \smsingc k U
          \R$ we have $\smear_M(c) \in \mhch k M$.
    \item The map $\smear_M \colon \smsingc * U \R \longrightarrow
          \mhch * M$ is a chain map.
  \end{enumerate}
\end{lem}
\begin{bew}
  1. Of course, we need only show this assertion for smooth singular
     simplices~$\sigma \in \maps[norm]{\ssim k}U$. According to 
     Lemma~\ref{tov}, $h_M$ is a signed measure of finite total
     variation because $h_M \geq 0$ and 
     \[ h_M\bigr( \pi_1(M) \setminus \isomp(U) \bigr) = 1. \] 
     Furthermore, $\pi_1(M)\setminus\isomp(U)$
     is a compact determination set of~$h_M$. Now 
     Lemma~\ref{mhabschluss} yields
     \[ \smear_M(\sigma) = h_M^{F_\sigma} \in \mhch k M  \]
     since $F_\sigma$ is, due to Lemma~\ref{intcont}, continuous.

  2. Let $k \in \N$, $j \in \{0, \dots, k+1\}$, and let $\sigma \in
     \maps[norm]{\ssim{k+1}}U$ be a smooth singular $(k+1)$-simplex. 
     By linearity, it is sufficient to show 
     \[ \smear_M(\face{j}{\sigma}) 
        = \bou_j\bigl( \smear_M(\sigma)\bigr)\]
     where $\face j \sigma$ stands for the $j$-th face of $\sigma$.
     By construction, 
     \begin{align*}
       \smear_M(\face j\sigma)
         & = h_M^{F_{\face j \sigma}} \\
         & = h_M^{\bigl( (\pi_1(M))\cdot f \mapsto p_M \circ f 
                                                       \circ \face j\sigma
                  \bigr)} \\
         & = \bigl( h_M^{F_\sigma}\bigr)^{(\tau \mapsto \tau \circ\bou_j)} \\
         & = \bou_j \bigl( h_M^{F_\sigma}\bigr) \\
         & = \bou_j\bigl(\smear_M(\sigma)\bigr).
             \qedhere
    \end{align*} 
\end{bew}

To pull back the smearing operation to smooth singular simplices
on~$N$ via $p_N$-lifts, we need the right invariance of the 
measure~$h_M$ to guarantee well-definedness:

\begin{lem}
  Let $k \in \N$ and $\sigma \in \maps[norm]{\ssim k}N$.
  \begin{enumerate}
    \item Then there is a $p_N$-lift of $\sigma$ lying
          in~$\maps[norm]{\ssim k} U$. In fact, each continuous 
          $p_N$-lift of~$\sigma$ is smooth.
    \item If $\sigma_1$, $\sigma_2 \in \maps[norm]{\ssim k}U$ are two 
          $p_N$-lifts of~$\sigma$, then
          \[ h_M^{F_{\sigma_1}} = h_M^{F_{\sigma_2}}.\]
  \end{enumerate}
\end{lem}
\begin{bew}
  1. Since $\ssim k$ is simply connected, there is a continuous
     $p_N$-lift 
     \[ \widetilde\sigma \colon \ssim k \longrightarrow U \]
     of~$\sigma$. Noting that the covering projection $p_N \colon U
     \longrightarrow N$ is a local diffeomorphism (even a local
     isometry), we deduce from the commutative diagram
     \begin{align*}
       \xymatrix@=3em{
            & {U}   \ar[d]^{p_N} \\
          {\ssim k} \ar[r]_{\sigma} \ar[ur]^{\widetilde\sigma} 
            & {N}
       }
     \end{align*}
     that $\widetilde\sigma$ also must be smooth (smoothness is a
     local property). 

  2. Since $\ssim k$ is connected, two continuous $p_N$-lifts
     of~$\sigma$ only differ by some element in~$\pi_1(N)$. 
     As already mentioned in Corollary~\ref{quotcomp}, the fundamental
     group~$\pi_1(N)$ acts by orientation preserving isometries on the universal
     cover~$U$. Hence there is a $g \in \isomp(U)$ such that
     \[ \sigma_1 = g \circ \sigma_2.\]

     Writing $r_g \colon \pi_1(M)\setminus\isomp(U) \longrightarrow
     \pi_1(M)\setminus\isomp(U)$ for right multiplication by~$g$, we get
     $F_{\sigma_1} = F_{\sigma_2} \circ r_g.$
     Thus,
     \[ h_M^{F_{\sigma_1}} 
                     = h_M^{F_{\sigma_2} \circ r_g}
                     = \bigl( h_M^{r_g}\bigr)^{F_{\sigma_2}}.\]
     Finally, the right invariance of~$h_M$ enters the scene: Since
     $h_M$ is right invariant, we have
     \[ h_M^{r_g} = h_M,\]
     which completes the proof. 
\end{bew}

Hence we can pull back $\smear_M$ to smooth singular chains on~$N$: 

\begin{defi}
  For each $k \in \N$ and each smooth singular simplex 
  $\sigma \in \maps[norm]{\ssim k}N$ we define its \defin{smeared
  chain}\dindex{smearing}\dindex{chain!smeared} by
  \[ \smear_{N,M}(\sigma) := \smear_M(\widetilde\sigma)  \in \mhch k
     M, \]%
  \nindex{smearNM}{$\smear_{N,M}$}{smearing}%
  where $\widetilde\sigma \in \maps[norm]{\ssim k}U$ is some $p_N$-lift
  of~$\sigma$. Again, the notation $\smear_{N,M}$ will also be used
  for its linear extension $\smsingc * N \R \longrightarrow \mhch * M$.
\end{defi}

\begin{lem}\tindex{smearing}
  The smearing map
  \[ \smear_{N,M} \colon \smsingc * N \R \longrightarrow \mhch * M\]
  is a chain map.
\end{lem}
\begin{bew}
  Let $k \in \N$, $j \in \{ 0, \dots, k+1\}$, and let $\sigma \in
  \maps[norm]{\ssim {k+1}}N$. If $\widetilde \sigma$ is a $p_N$-lift,
  then $\face j {\widetilde\sigma}$ is obviously a $p_N$-lift
  of~$\face j \sigma$. Hence we conclude from Lemma~\ref{smuch}
  \begin{align*}
    \smear_{N,M}(\face j \sigma) 
      & = \smear_M(\face j {\widetilde\sigma}) \\
      & = \bou_j\bigl( \smear_M(\widetilde \sigma)\bigr) \\
      & = \bou_j\bigl(\smear_{N,M}(\sigma) \bigr).
  \end{align*}
  Thus $\smear_{N,M}$ is a chain map.
\end{bew}

Furthermore, smearing does not spoil the norm: 

\begin{lem}\label{smnorm}
  For all $k \in \N$ and all $c \in \smsingc k N \R$ 
  \[ \tov{\smear_{N,M}(c)} \leq \smlone c.\]
\end{lem}
\begin{bew}
  Since the measure~$h_M$ is positive, so is
  $\smear_{N,M}(\sigma)$ for all smooth singular simplices 
  $\sigma\in\maps{\ssim k}N$. This implies 
  \begin{align*} 
          \tov{\smear_{N,M}(\sigma)} 
        & = \smear_{N,M}(\sigma) \bigl(\maps[norm]{\ssim k}U\bigr)\\
        & = h_M\bigl( \pi_1(M)\setminus\isomp(U)
               \bigr) \\
        & = 1. 
  \end{align*}
  Now the general case is an immediate consequence of the triangle
  inequality for the total variation (compare Lemma~\ref{tovnorm}).
\end{bew}

The integration operation introduced in Section~\ref{integration} makes 
it possible to understand the effect $\smear_{N,M}$ has on homology:


\begin{satz}\label{smearthm}\tindex{smearing}
  Smearing induces multiplication by $\vol(N) / \vol(M)$ on the 
  top homology group. More precisely: for all cycles $z \in \smsingc n N \R$
  with $[z] = \smfcl N$ in $\hsmsingc n N \R$, 
  \[ \bigl[  \smear_{N,M}(z) \bigr]
          = \frac{\vol(N)}{\vol(M)}
            \cdot \mhfcl M \in \hmh n M. \]
\end{satz}

The proof basically relies on the following calculation:

\begin{lem}\label{intsm}
  For all $\sigma \in \maps[norm]{\ssim k}{N}$ the following integrals are 
  equal:
  \[ \int_{\smear_{N,M}(\sigma)} \vol_M = \int_\sigma \vol_N.\]
\end{lem}
\begin{bew}[Proof (of Lemma~\ref{intsm}).]
  Let $\ucov \sigma$ be some $p_N$-lift of $\sigma$. By definition, the integral can  
  be computed as follows: 
  \begin{align*}
       \int_{\smear_{N,M} (\sigma)} \vol_M
   & = \int_{\maps[norm]{\ssim k}{M}} \int_\tau \vol_M 
            \,d\bigl(\smear_{N,M}(\sigma)\bigr)(\tau) \\
   & = \int_{\maps[norm]{\ssim k}{M}} \int_\tau \vol_M 
            \,d\bigl(h_M^{F_{\ucov \sigma}}\bigr)(\tau).
  \end{align*}
  Using the transformation formula for measures, we get
  \begin{align*}
       \int_{\smear_{N,M} (\sigma)} \vol_M
   & = \int_{\pi_1(M)\setminus \isomp(U)} 
            \int_{F_{\ucov\sigma}(\pi_1(M) \cdot f)} \vol_M 
            \,d h_M (\pi_1(M) \cdot f) \\
   & = \int_{\pi_1(M)\setminus \isomp(U)} 
            \int_{p_M \circ f \circ \ucov \sigma} \vol_M 
            \,d h_M (\pi_1(M) \cdot f) \\
   & = \int_{\pi_1(M)\setminus \isomp(U)} 
            \int_{\ucov \sigma} f^*p_M^*\vol_M 
            \,d h_M (\pi_1(M) \cdot f). 
  \end{align*}
  By definition, $p_M$ and $p_N$ are orientation preserving local 
  isometries. 
  Since orientation preserving local isometries between Riemannian manifolds 
  pull back the volume form to the volume form, we conclude
  \begin{align*}
       \int_{\smear_{N,M} (\sigma)} \vol_M
   & = \int_{\pi_1(M)\setminus \isomp(U)} 
            \int_{\ucov \sigma} \vol_U 
            \,d h_M (\pi_1(M) \cdot f) \\
   & =  \int_{\pi_1(M)\setminus \isomp(U)} 
            \int_{\ucov \sigma} p_N^*\vol_N 
            \,d h_M \\
   & =  \int_{\pi_1(M)\setminus \isomp(U)}
            \,d h_M
        \cdot \int_{\ucov \sigma} p_N^*\vol_N \\
   & = 1 \cdot \int_{p_N \circ \ucov \sigma} \vol_N \\
   & = \int_{\sigma} \vol_N. \qedhere
  \end{align*}
\end{bew}



\begin{bew}[Proof (of Theorem~\ref{smearthm}).]
  Lemma~\ref{intsm} and Lemma~\ref{svvol} imply 
  \[ \int_{\smear_{N,M} (z)} \vol_M = \int_z \vol_N = \vol(N). \]
  Application of Theorem~\ref{mhdr} therefore results in
  \[    \bigl[ \smear_{N,M}(z) \bigr] 
     =  \frac{\vol(N)}{\vol(M)} \cdot \mhfcl{M} \in \hmh n M.
     \qedhere
  \]
\end{bew}

\begin{bem}
  It is possible to extend the smearing chain map~$\smear_{N,M}$ to all measure
  chains: 
  \begin{align*}
     \mhch k N & \longrightarrow \mhch k M \\
           \mu & \longmapsto 
                 \biggl( A \mapsto
                         \int_{\maps{\ssim k} N} 
                         h_M^{F_{\ucov \sigma}} (A) 
                         \; d\mu(\sigma)
                 \biggr).
  \end{align*}
  This smearing map also does not increase the norm and induces
  multiplication by~$\vol(N) / \vol(M)$ on top homology. This leads to
  an intrinsic proof for the proportionality principle of the
  ``simplicial volume'' defined via measure homology. 
\end{bem}

\newpage

% ----------------------------------------------------------------
% ----------------------------------------------------------------

\section{Proof of the proportionality principle}\label{pproof}

Combining the techniques from Section~\ref{smearing} and the isometric
isomorphism of Section~\ref{isometricisom}, we are now in state to do
the final step in the proof of the proportionality principle of
simplicial volume:

\tindex{proportionality principle}%
\begin{bew}[Proof (of Theorem~\ref{propprinc}).]
  Suppose $z \in \smsingc n N \R$ is a cycle which represents the 
  (smooth) fundamental class~$\smfcl N$.
  Theorem~\ref{smearthm} shows that $\smear_{N,M}$ induces the
  factor~$\vol(N)/\vol(M)$ on top homology, i.e., 
  \[ \bigl[ \smear_{N,M}(z)\bigr]
   =  \frac{\vol(N)}{\vol(M)} \cdot \mhfcl{M}. \]
  Since we can compute the simplicial volume via measure homology
  (Corollary~\ref{isomisomc}) and since smearing does not increase the
  norm (Lemma~\ref{smnorm}), we obtain
  \begin{align*}
    \sv M & =    \mhnorm{\mhfcl M} \\
          & \leq \frac{\vol(M)}{\vol(N)} \cdot 
                 \tov{\smear_{N,M}(z)} \\
          & \leq \frac{\vol(M)}{\vol(N)} \cdot
                 \smlone z.
  \end{align*}
  Now taking the infimum over all (smooth) representatives of~$\smfcl N$ 
  gives the estimate
  \[ \sv M \leq \frac{\vol(M)}{\vol(N)} \cdot \smlone{\smfcl N} 
           = \frac{\vol(M)}{\vol(N)} \cdot \sv N \]
  because $\smlone{\smfcl N}  =\sv N$ by Corollary~\ref{isomisomsm}.
  Swapping the r\^oles of the manifolds 
  $M$ and $N$ shows the reverse inequality
  \[ \sv N \leq \frac{\vol(N)}{\vol(M)} \cdot \sv M \]
  and hence proves the proportionality principle.
\end{bew}


% -----------------------------------------------------------
% -----------------------------------------------------------

\section{Applications of the proportionality principle}\label{ppapp}%
\tindex{proportionality principle!applications}

Using Gromov's upper bound for the simplicial volume in terms of the
minimal volume (Theorem~\ref{minvolsv}), we already have seen that the
simplicial volume of flat oriented closed connected manifolds must
vanish. We will give now a proof of this fact based on the
proportionality principle instead of applying Gromov's sophisticated 
estimate:

\begin{koro}\label{flat}\tindex{simplicial volume!of flat manifolds}%
  \dindex{flat manifold}%
  If $M$ is an oriented closed connected flat Riemannian manifold
  (i.e., it is locally isometric to Euclidean space), then
  \[ \sv{M}  =  0.\]
\end{koro}
\begin{bew}

  For $c \in \R_{> 0}$ we consider the Riemannian manifold $M_c$ which
  is obtained from $M$ by scaling its Riemannian metric by the
  factor~$c^2$ (i.e., all lengths are stretched by the factor~$c$).
  Then $M$ and $M_c$ are homeomorphic manifolds (in particular,
  $\sv{M} = \sv{M_c}$) and
  \[ \vol(M_c) = c^n \cdot \vol(M) \]
  where $n$ is the dimension of $M$. Moreover, $M_c$ is also flat.
  Therefore both universal covers $\ucov{M}$ and~$\ucov{M_c}$ are
  isometric to $\R^n$ \cite[Theorem~11.12]{lee}.  Thus the
  proportionality principle yields
  \[ \frac{\sv{M}}{\vol(M)} 
   = \frac{\sv{M_c}}{\vol(M_c)}
   = \frac{\sv{M}}{c^n \cdot \vol(M)}. \]
  Since this holds for all $c \in \R_{>0}$, clearly $\sv{M} = 0$ follows.
\end{bew}

Since the simplicial volume is a homotopy invariant, we can deduce from
the proportionality principle that the Riemannian volume in some
special cases is homotopy invariant:

\begin{qedkoro}
  If $M$ and $N$ are homotopy equivalent oriented closed connected
  Riemannian manifolds with isometrically isomorphic universal
  Riemannian coverings and $\sv M \neq 0$, then
  \[ \vol(M) = \vol(N). \qedhere \]
\end{qedkoro}
\sindex{topology!rocks"!}{topology}%

simvol.tex
% main:: diploma_main.tex
% 

\section{Singular (co)homology refreshed}

Before defining the simplicial volume, we recall the basic definitions
of singular (co)homology and introduce the corresponding notation
which will be used throughout this thesis. The key objects of singular
(co)homology are simplices:

\begin{defi}
  For $k \in \N$ the set 
  \[ \ssim{k} := 
        \biggl\{ 
        (x_0, \dots, x_k) \in \R_{\geq 0}^{k+1}
        \bigg|\; \sum_{j=0}^k x_j = 1
        \biggr\}\]
  is called the $k$\defin{-dimensional standard simplex}. 
  \dindex{standard simplex}%
  \dindex{simplex!standard}%
  \nindex{Delta}{$\ssim k$}{standard $k$-simplex}%
  For $j \in \{0, \dots, k\}$ we let 
  \begin{align*}
     \bou_j \colon \ssim{k-1} & \longrightarrow \ssim{k} \\
        (x_0, \dots, x_{k-1}) & \longmapsto
                                (x_0, \dots, x_{j-1},0,x_j, \dots, x_{k-1})
  \end{align*}
  be the inclusion of the $j$\defin{-th face of} $\ssim{k}.$
  \nindex{dj}{$\bou_j$}{$j$-th face of a standard simplex}
  \dindex{face}
\end{defi}

To be precise, the face mappings $\bou_j$ also depend on $k,$ but we 
omit $k$ in the notation since it can easily be inferred from the context.

The main idea of the singular theory is to examine all continuous maps
from the standard simplices to a topological space to detect the
``holes'' of this space, i.e., to measure which ``cycles'' are no
``boundaries''. 

\begin{defi}
  Let $X$ be a topological space and $k \in \N.$
  \begin{itemize}
     \item The set $\ssing{k}{X}$ of \defin{singular }$k$\defin{-simplices} is
           the set of all continuous functions $\ssim{k}
           \longrightarrow X.$%
           \nindex{S}{$\ssing k X$}{set of singular $k$-simplices in~$X$}%
           \dindex{singular simplex}%
           \dindex{simplex!singular}
           If $\sigma \in \ssing{k}{X}$ and $j \in
           \{0, \dots, k\},$ then $\face{j}{\sigma}$ is called the
           $j$\defin{-th face of }$\sigma.$ 
           \nindex{dj}{$\bou_j$}{$j$-th face of a singular simplex}
           \dindex{face}
     \item Let $R$ be a commutative ring with unit. Then the set
           $\csingc{k}{X}{R}$ of 
           \defin{singular }$k$\defin{-chains of }$X$\defin{ with
           coefficients in }$R$ is the free $R$-module generated by
           $\ssing{k}{X}.$%
           \nindex{C}{$\csingc * X R$}{singular chains of~$X$ with coefficients
           in~$R$}%
           \dindex{singular chain}%
           \dindex{chain!singular}
     \item The \defin{boundary operator} $\bou \colon
           \csingc{k+1}{X}{R} \longrightarrow \csingc{k}{X}{R}$ is the 
           $R$-module homomorphism given on the generators $\sigma \in
           \ssing{k+1}{X}$ by
           \[ \bou(\sigma) := \sum_{j=0}^{k+1} (-1)^j \cdot \face{j}{\sigma}.\]
           \nindex{d}{$\bou$}{boundary operator}%
           \dindex{boundary operator}%
           The elements of $\im \bou$ are called -- as geometry
           suggests 
           -- \defin{boundaries}, and the elements of $\ker
           \bou$ are called \defin{cycles}.
           \dindex{boundary}
           \dindex{cycle}
     \item We define the set $\cosingc{k}{X}{R}$ of 
           \defin{singular }$k$\defin{-cochains of }$X$\defin{ with 
           coefficients in }$R$ as the algebraic dual of
           $\csingc{k}{X}{R},$ i.e.,%
           \nindex{Co}{$\cosingc * X R$}{singular cochains of~$X$ with
           coefficients in~$R$}%
           \dindex{singular cochain}%
           \dindex{cochain!singular}
           \[ \cosingc{k}{X}{R} := \Hom_{\Z}\bigl( \csingc{k}{X}\Z,
                                                 R\bigr)
                                 = \Hom_R\bigl(\csingc{k}{X}{R},R\bigr)\]
     \item The boundary operator $\bou \colon
           \csingc{k+1}{X}{R} \longrightarrow \csingc{k}{X}{R}$
           induces the \defin{co\-boundary operator} $\cobou \colon
           \cosingc{k}{X}{R} \longrightarrow \cosingc{k+1}{X}{R}$ as
           follows:%
           \dindex{coboundary operator}%
           \nindex{delta}{$\cobou$}{coboundary operator\\}
           \begin{align*}
              \cobou \colon \cosingc{k}{X}{R} 
                   & \longrightarrow \cosingc{k+1}{X}{R} \\
                 f & \longmapsto \bigl(c 
                     \mapsto (-1)^{k+1} \cdot f(\bou c)\bigr).
           \end{align*}
           The elements of $\im \cobou$ are called
           \defin{coboundaries} and the elements of $\ker \cobou$ are
           called \defin{cocycles}.%
           \dindex{coboundary}%
           \dindex{cocycle}
  \end{itemize}
  In the special case $R = \Z$, we use the abbreviations $\csing{k}{X}
  := \csingc{k}{X}{\Z}$ and $\cosing k X := \cosingc k X \Z$.
  \nindex{C}{$\csing * X$}{$=\csingc * X \Z$}
  \nindex{Co}{$\cosing * X$}{$= \cosingc * X \Z$}
\end{defi}

A lengthy calculation shows that in the situation of the above
definition the equality $\bou \circ \bou = 0$ holds (i.e., the
singular chains form a chain complex). Therefore $\cobou
\circ \cobou = 0,$ too. Hence the following definition makes sense:

\begin{defi}
  Let $X$ be a topological space, $k \in \N,$ and let $R$ be a commutative
  ring with unit. 
  \begin{itemize}
     \item The $R$-module
           \[ \hsingc{k}{X}{R} := 
              \frac{\ker \bigl(\bou\colon \csingc{k}{X}{R}
                               \rightarrow \csingc{k-1}{X}{R}
                         \bigr)}
                   {\im  \bigl(\bou\colon \csingc{k+1}{X}{R}
                               \rightarrow \csingc{k}{X}{R}
                         \bigr)}\]
           is called $k$\defin{-th singular homology of }$X$\defin{
           with coefficients in }$R.$%
           \nindex{H}{$\hsingc * X R$}{singular homology of~$X$ with coefficients
           in~$R$}%
           \dindex{singular homology}%
           \dindex{homology!singular}
     \item The $R$-module
           \[ \cohsingc{k}{X}{R}:= 
              \frac{\ker \bigl(\cobou\colon \cosingc{k}{X}{R}
                               \rightarrow \cosingc{k+1}{X}{R}
                         \bigr)}
                   {\im  \bigl(\cobou\colon \cosingc{k-1}{X}{R}
                               \rightarrow \cosingc{k}{X}{R}
                         \bigr)}\]
           is called $k$\defin{-th singular cohomology of }$X$\defin{
           with coefficients in }$R.$%  
           \nindex{Ho}{$\cohsingc * X R$}{singular cohomology of~$X$ with
           coefficients in~$R$}%
           \dindex{singular cohomology}%
           \dindex{cohomology!singular}
  \end{itemize}
  For the sake of brevity we write
  \begin{align*}
     \hsing{k}{X}   & := \hsingc{k}{X}{\Z},\\
     \cohsing{k}{X} & := \cohsingc{k}{X}{\Z}. \qedhere
  \end{align*}
  \nindex{H}{$\hsing * X$}{$=\hsingc * X \Z$}
  \nindex{Ho}{$\cohsing * X$}{$=\cohsing * X \Z$}
\end{defi}

For convenience, we introduce the following convention:

\begin{defi}
  Let $M$ be an $R$-module, $B \subset M$, and $(a_b)_{b \in B}
  \subset R$. When we write $\fsum_{b
  \in B} a_b \cdot b$, we implicitly assume that all but a finite
  number of the coefficients~$(a_b)_{b \in B}$ are~0 unless stated
  otherwise. Additionally, whenever we write something like 
  \begin{center} 
   ``Let $\fsum_{\sigma \in \ssing k X} a_\sigma \cdot \sigma \in
            \csingc k X R$.'' 
  \end{center}
  it is understood that all~$a_\sigma$ are elements of~$R$. 
\end{defi}

\begin{defi}
  Let $f \colon X \longrightarrow Y$ be a continuous map and let $R$
  be a commutative ring with unit. The map~$f$ induces chain maps 
  \begin{align*}
    \csingc k f R \colon \csingc k X R & \longmapsto \csingc k Y R \\
      \fsum_{\sigma \in \ssing k X} a_\sigma \cdot \sigma
     & \longmapsto \fsum_{\sigma \in \ssing k X} a_\sigma \cdot f\circ\sigma
  \end{align*}
  \nindex{C}{$\csingc * f R$}{homomorphism induced by~$f$}
  and
  \begin{align*}
    \cosingc k f R \colon \cosingc k Y R & \longmapsto \cosingc k X R\\
      g & \longmapsto g \circ \csingc k f R.
  \end{align*}%
  \nindex{Co}{$\cosingc * f R$}{homomorphism induced by~$f$}%
  The corresponding maps on (co)homology are denoted by $\hsingc * f
  R$ and $\cohsingc * f R$ respectively. This turns singular
  (co)homology into a functor.
  \nindex{H}{$\hsingc * f R$}{homomorphism induced by~$f$}%
  \nindex{Ho}{$\cohsingc * f R$}{homomorphism induced by~$f$}%
  \nindex{H}{$\hsing * f$}{$=\hsingc * f \Z$}%
  \nindex{Ho}{$\cohsing * f$}{$=\cohsingc * f \Z$}%
\end{defi}


More details on singular (co)homology can be found in any textbook on
algebraic topology.


% -------------------------------------------------------------------
\section{Definition of the simplicial volume}\label{svdefsec}

We will first introduce a norm on the singular chain complex with real
coefficients, the $\ell^1$-norm. This norm  induces a seminorm on 
singular homology.
The simplicial volume of an oriented closed connected manifold is then
defined as the seminorm of the fundamental homology class.

\begin{defi}
  Let $X$ be a topological space, and $k \in \N$. For all singular
  chains $a = \fsum_{\sigma
  \in \ssing{k}{X}} a_\sigma \cdot \sigma \in \csingc{k}{X}{\R}$  
  we define 
  \nindex{$h}{$\lone{\,\cdot\,}$}{$\ell^1$-norm}%
  \[ \lone{a} := \fsum_{\sigma\in\ssing{k}{X}} |a_{\sigma}|.\qedhere\]
\end{defi}

Since the sum is finite, this norm is well-defined and turns the singular
chain group $\csingc{k}{X}{\R}$ into a normed real vector space.

\begin{defi}
 Let $X$ be a topological space, $k\in\N$ and $\alpha \in
 \hsingc{k}{X}{\R}$. 
 Then
 $\lone{\, \cdot \,}$ induces a seminorm as follows:
 \nindex{$i}{$\lone{\,\cdot\,}$}{seminorm}%
 \dindex{seminorm}
 \[ \lone{\alpha} := \inf 
     \bigl\{ \lone{a} \;\big|\;
              a \in \csingc{k}{X}{\R}, \bou(a)=0, [a] = \alpha 
     \bigr\}. \qedhere\]
\end{defi}

In general, this is not a norm since the set of boundaries does not need to
be closed in the $\ell^1$-norm. 
Geometrically, this seminorm measures how efficiently a homology class can be
represented by singular chains.

The fundamental homology class of an $n$-dimensional oriented closed
connected manifold $M$ is the generator of~$\hsing{n}{M} \cong \Z$
which is compatible with the orientation of~$M$
\cite[Theorem~XIV~2.2]{massey}.

\begin{defi}
  Let $M$ be an oriented closed connected $n$-dimensional manifold. We
  will denote the corresponding fundamental homology class of $M$ by
  $\fcl{M} \in \hsing{n}{M}.$% 
  \dindex{fundamental class!in singular homology}%
  \nindex{$t}{$\fcl M$}{fundamental class}%
  \nindex{$u}{$\fclr M$}{fundamental class with real coefficients} 
  Moreover, we write $\fclr{M}$ for the image of $\fcl{M}$ under the
  change of coefficients homomorphism
  $ \hsingc{n}{M}{\Z} \longrightarrow \hsingc{n}{M}{\R}.$
\end{defi}

\begin{defi}%
  \nindex{$m}{$\sv M$}{simplicial volume}%
  \dindex{simplicial volume}%
  \sindex{volume!simplicial}{simplicial~volume}
  The \defin{simplicial volume} of an oriented closed connected
  manifold $M$ is defined as
  \[ \sv{M} := \lone[big]{\fclr{M}}.\qedhere\]
\end{defi}

Since $\lone{\,\cdot\,}$ is not affected by multiplication with~$-1$,
the simplicial volume is independent of the chosen orientation.

A good geometric description of $\fcl{M}$ can be given in terms of
triangulations 
\cite[page~338f]{bredon}: if $M$ is triangulable and $\sigma_1, \dots,
\sigma_m \colon \ssim{n} \longrightarrow M$ is a triangulation of $M$,
then $\sum_{j=1}^{m} \sigma_j$ represents~$\fcl{M}$.  
Hence the simplicial volume measures in some sense the complexity of~$M$.

Of course, it is also possible to use other seminorms on singular
homology (to get different ``simplicial volumes'') or to generalise
the above definition to a larger class of spaces. We will sketch such
extensions in Section~\ref{svgen}.

The following lemma indicates why the simplicial volume is akin to 
a volume (for the simplicial volume, the simplices are all given
the weight~$1$ instead of the integral expression, and only the absolute values
of the coefficients are considered).

\begin{lem}\label{svvol}\tindex{integration!of smooth singular simplices}%
  \tindex{smooth singular simplex!integration of}%
  \tindex{singular simplex!integration of}%
  \tindex{integration!and volume}
  Let $M$ be an oriented closed connected Riemannian manifold of
  dimension $n.$ If $\fsum_{\sigma \in \ssing{n}{M}} a_\sigma\cdot
  \sigma \in \csingc n M \R$ is a cycle representing $\fclr M$ consisting of
  smooth singular simplices, then 
  \[ \vol(M) = \fsum_{\sigma\in\ssing{n}{M}} a_\sigma \cdot
                      \int_\sigma \vol_M.\]
  Here $\vol_M$ denotes the volume form of~$M$ and the integral is
  explained in Definition~\ref{defint}.
\end{lem}
\begin{bew}
  Since $M$ is smooth, there is a triangulation~$\sigma_1, \dots,
  \sigma_m \colon \ssim n \longrightarrow M$ \cite{whitehead}. 
  Then one shows that the lemma is true for the cycle~$\sum_{j=1}^m
  \sigma_j$ (which is intuitively clear) and uses the fact that the
  integral only depends on the represented homology class
  \cite[Exercise~16.2 and Theorem~16.10]{leesmooth}. 
\end{bew}

Stronger connections -- such as the proportionality principle --
between the simplicial volume and other volumes are discussed in
Subsection~\ref{ovol} and Chapter~\ref{chpropprinc}.

As a first example we will give a direct calculation for~$\sv[norm]{S^1}$.
There are much more sophisticated techniques for
computing $\sv[norm]{S^1}$, but the following proof emphasises
geometric intuition. 

\begin{satz}\label{firsthom}% Lueck, Lemma 14.14
  Let $X$ be an arcwise connected topological space. Then $\lone\alpha
  =0$ holds for all~$\alpha \in \hsingc 1 X \R$.
\end{satz}
\begin{bew}
  The universal coefficient theorem yields
  \[ \hsing{1}{X} \otimes \R \cong \hsingc{1}{X}{\R}.\]
  In other words: if $\alpha \in \hsingc{1}{X}\R,$ then there exist $k
  \in \N$, $r_1, \dots, r_k \in \R$, and $\alpha_1, \dots, \alpha_k \in
  \hsing{1}{X}$ such that $\alpha$ corresponds to 
  \[ \sum_{j=1}^k r_j \cdot \alpha_j.\] 
  By the Hurewicz theorem each of the $\alpha_j$ can be represented by
  a continuous function $\sigma_j \colon S^1 \longrightarrow X.$ For each
  $j \in\{1,\dots,k\}$ and each $d \in \N_{\geq 1}$ we write
  \smash{$\sigma^{(d)}_j$} for the composition (``wrapping $\sigma_j$
  around the circle $d$ times'')
  \[ \sigma_j \circ (z \mapsto z^d) \colon S^1 \longrightarrow X.\]
  Now the Hurewicz theorem tells us that
  $ [ \sigma^{(d)}_j] = d \cdot [\sigma_j]$
  holds in $\hsing{1}{X}$ (and thus also in~$\hsingc 1 X \R$). 
  In particular,
  \begin{align*}
    \lone{\alpha} & \leq \lone[bigg]{\sum_{j=1}^k r_j \cdot
                               \frac{\sigma_j^{(d)}}{d}} 
                    \leq \frac{1}{d} \cdot \sum_{j=1}^k |r_j|.  
  \end{align*}
  Since this is true for any $d \in \N_{\geq 1},$ we get
  $\lone{\alpha} = 0$.
\end{bew}
\begin{qedkoro}%
  \tindex{simplicial volume!of S^1@of~$S^1$}
  In particular, $\sv[norm]{S^1} = 0$.
\end{qedkoro}

This corollary also implies that the seminorm~$\lone{\,\cdot\,}$ 
on~$\hsingc 1 {S^1} \R$ is not a norm.

It is more difficult to show that there are manifolds with non-zero 
simplicial volume. For example, the simplicial volume of hyperbolic
manifolds does not vanish (see Theorem~\ref{hypp}).


% -------------------------------------------------------------------
\section{A zoo of properties}\label{svpropsec}

In the first subsection, some elementary properties, such as homotopy
invariance and multiplicativity of the simplicial volume are deduced
from the definition. 
We then give a collection of more sophisticated properties, based on
the comparison with other volumes and on the theory of bounded
cohomology. 
This section ends with a short (and rather incomplete) discussion of
realisability. 

\subsection{Multiplicativity}

One of the main characteristics of the simplicial volume is that it is  
multiplicative with respect to the degree of maps, which enables us to compute
the simplicial volume in some special cases.

We start with a simple  -- but fundamental -- observation. Gromov
calls the described property \defin{functoriality}  
\cite[5.34]{gromovb}.
\dindex{functoriality}

\begin{lem}\label{lonenfunc}
  Let $f \colon X \longrightarrow Y$ be a continuous map of topological
  spaces, $k \in \N$ and $\alpha \in H_k(X,\R)$. Then we get for the
  induced seminorm
  \[ \lone{\hsingc{k}{f}{\R}(\alpha)} \leq \lone{\alpha}.\]
\end{lem}
\begin{proof}
  Let $\fsum_{\sigma \in \ssing{k}{X}} a_\sigma \cdot \sigma \in
  \csingc{k}{X}{\R}$ 
  be a cycle representing $\alpha.$ Then the class
  $\hsingc{k}{f}{\R}(\alpha)$ is represented by
  $\fsum_{\sigma \in \ssing{k}{X}} a_\sigma \cdot f \circ \sigma.$ Therefore
  \begin{align*}%
           \lone{\hsingc{k}{f}{\R}(\alpha)}
    & =    \lone[bigg]{\biggl[\fsum_{\sigma \in \ssing{k}{X}} 
                 a_\sigma \cdot f \circ \sigma\biggr]}
      \leq \lone[bigg]{\fsum_{\sigma \in \ssing{k}{X}} 
                 a_\sigma \cdot f \circ \sigma} \\
    & \leq \fsum_{\sigma \in \ssing{k}{X}}
                 | a_\sigma | 
      =    \lone[bigg]{\fsum_{\sigma \in \ssing{k}{X}} 
                 a_\sigma \cdot \sigma}. 
  \end{align*} 
  Taking the infimum over all cycles representing $\alpha$ yields the
  desired inequality
  $\lone{\hsingc k f \R(\alpha)} \leq \lone{\alpha}.$
\end{proof}

Multiplicativity (with respect to the degree of maps) is now a direct 
consequence of this lemma. 

\begin{defi}%
  \nindex{deg}{$\deg$}{degree of a map}%
  \dindex{degree}
  Let $M$ and $N$ be oriented closed connected $n$-dimensional manifolds
  and let $\fcl{M} \in \hsing{n}{M}, \fcl{N} \in \hsing{n}{N}$ be
  their fundamental homology classes. Then the \defin{degree} of a
  continuous map $f \colon M \longrightarrow N$
  is defined as the integer~$\deg(f)$ satisfying 
  \[ \hsing{n}{f}\bigl(\fcl{M}\bigr)
     = \deg(f) \cdot \fcl{N}. \qedhere\]
\end{defi}

\begin{bem}
  In the situation of the definition,
  \[ \hsingc{n}{f}{\R}\bigl(\fclr{M}\bigr) 
     = \deg(f) \cdot \fclr{N}\]
  also holds (by the universal coefficient theorem).
\end{bem}

\begin{lem}\label{mult}\tindex{multiplicativity}
  Let $f \colon M \longrightarrow N$ be a map of oriented closed
  connected manifolds of the same dimension. Then
  \[ \sv{M} \geq |\deg(f)| \cdot \sv{N}.\]
\end{lem}
\begin{proof}
  Using  the previous lemma and the above remark we conclude that  
  \begin{align*}
    \sv{M} & =    \lone[big]{\fclr{M}} \\
           & \geq \lone[big]{\hsingc{*}{f}{\R}(\fclr{M})} \\
           & =    \lone[big]{\deg(f) \cdot \fclr{N}} \\
           & =    |\deg(f)| \cdot \sv{N}. \qedhere
  \end{align*}
\end{proof}

\begin{qedkoro}\label{svhi}
  \tindex{homotopy invariance!of simplicial volume}
  \tindex{simplicial volume!homotopy invariance of}
  Since homotopy equivalences between oriented closed connected manifolds
  are maps of degree $1$ or $-1$, the simplicial
  volume is a homotopy invariant.
\end{qedkoro}

\begin{qedkoro}\label{svselfmap}
  \tindex{simplicial volume!of $S^n$}
  If $f \colon M \longrightarrow M$ is a self-map of an oriented
  closed connected manifold with $| \deg (f) | \geq 2,$ then 
  \[ \sv{M} = 0. \]
  In particular, $\sv[norm]{S^n} = 0$ for all $n\in\N_{\geq 1}$ and 
  $\sv[norm]{S^1 \times S^1} =0$. 
\end{qedkoro}


\begin{koro}\label{cov}
  \tindex{multiplicativity}
  Let $p \colon M \longrightarrow N$ be a $d$-sheeted covering of
  oriented closed connected manifolds. Then
  \[ \sv{M} = d \cdot \sv{N}. \]
\end{koro}
\begin{proof}
  \newcounter{acht}
  \setcounter{acht}{8}
  Let $n := \dim N.$ Since $p$ is a local homeomorphism, it also follows
  that $\dim M = n$. Of course, we may assume that $p$ is
  orientation preserving. Thus $\deg(p)$ is well-defined and \cite[Proposition
  \Roman{acht}.4.7]{dold}
  \[ \deg(p) = d.\]
  Hence, $\sv{M} \geq d \cdot \sv{N}.$ 
  As second step, we have to prove the ``$\leq$'' direction: 
  For $\sigma\in\ssing{n}{N}$, let $L(\sigma)$ be the set of singular
  simplices $\ucov{\sigma} \in \ssing{n}{M}$ with $p \circ \ucov\sigma
  = \sigma.$ Since $p$ is a $d$-sheeted covering and $\ssim{n}$ is
  arcwise connected and simply connected, the set $L(\sigma)$ contains
  precisely $d$ elements. If $a = \fsum_{\sigma \in \ssing{n}{N}} 
  a_\sigma \cdot \sigma \in \csingc{n}{N}{\R}$ represents $\fclr{N},$ 
  then
  \begin{align*}
    \ucov{a} & := \fsum_{\sigma \in \ssing{n}{N}} a_\sigma 
               \cdot \sum_{\ucov\sigma \in L(\sigma)} \ucov\sigma             
  \end{align*}
  represents $\fclr{M}$ because $\bou(\ucov a) = 0$,
  \begin{align*}
    \hsingc{n}{p}{\R}\bigl([\ucov a]\bigr) 
      & = \biggl[ \fsum_{\sigma\in\ssing{n}{N}} 
                  a_\sigma \cdot d \cdot \sigma \biggr]  
        = d \cdot \fclr{N} \\
      & = \hsingc{n}{p}{\R}\bigl(\fclr{M}\bigr), 
  \end{align*} 
  and $\deg(p) = d \neq 0$. Thus
  \begin{align*}
     \sv{M} & =    \lone[big]{\fclr{M}} 
              \leq \lone{\ucov{a}}
              \leq d \cdot \fsum_{\sigma\in\ssing{n}{N}} |a_\sigma| 
              = d \cdot \lone{a},
  \end{align*}
  which implies $\sv{M} \leq d \cdot \lone[big]{\fclr{N}} = d \cdot \sv{N}.$
\end{proof}

Using this result we can apply covering theory to obtain an upper
bound for the simplicial volume of oriented closed connected
surfaces \cite[page~217]{gromov}, \cite[Proposition~C.4.7]{bp}:

\begin{satz}\label{fg}
  \tindex{simplicial volume!of surfaces}
  Let $M$ be an oriented closed connected surface of genus
  $g \in \N_{\geq 2}.$ Then 
  \[ \sv{M} \leq 4g - 4.\]
\end{satz}

To prove this theorem we need the following (slightly weaker) inequality:

\begin{lem}\label{fg2}
  Let $g \in \N_{\geq 2}$ and let $F_g$%
  \nindex{Fg}{$F_g$}{oriented closed connected surface of genus~$g$} 
  be the oriented closed connected surface of genus~$g$. Then 
  \[ \sv{F_g} \leq 4g-2.\]
\end{lem}
\begin{bew}
  The surface $F_g$ can be constructed from a regular $4g$-gon by
  identifying certain edges \cite[\S7 of Chapter~I]{massey}. 
  This $4g$-gon can obviously be triangulated by $4g-2$ simplices:

  \begin{center}
    \includegraphics{fg.eps}
  \end{center}

  This triangulation induces a ``triangulation'' of $F_g$ consisting
  of $4g-2$
  simplices $\sigma_1, \dots, \sigma_{4g-2}.$ This is no triangulation
  of $F_g$ in the strict sense, but its second barycentric subdivision
  is. Hence we get \cite[page~338f]{bredon}
  that $ \sum_{j=1}^{4g-2} \sigma_j$ is a cycle representing
  $\fclr{F_g},$
  which yields the inequality 
  \[ \sv{F_g} \leq 4g-2. \qedhere\] 
\end{bew}

One can now combine the above inequality with a beautiful covering theory
argument to prove the theorem.

\begin{bew}[Proof (of Theorem \ref{fg}).]
  For each $d \in \N_{\geq 2}$ there is a $d$-sheeted covering 
  \[p_d \colon M_d \longrightarrow M\] 
  with connected covering space: we have
  \cite[Example~IV~5.3]{massey} 
  \begin{align*}
   \pi_1(F_g)^{\text{ab}} & \cong \Z^{2g},
  \end{align*}
  where $\pi_1(F_g)^{\text{ab}}$ is the abelianisation
  of~$\pi_1(F_g)$. Thus there is a surjective group homomorphism
  \[ \pi_1(M) \cong \pi_1(F_g) 
     \longrightarrow \pi_1(F_g)^{\text{ab}} \cong \Z^{2g}
     \longrightarrow \Z.\]
  The inverse image under this homomorphism of a subgroup of~$\Z$ 
  of index~$d$ is a subgroup of~$\pi_1(M)$ of index~$d.$ 
  Therefore $M$ admits a $d$-sheeted covering~$p_d \colon M_d
  \longrightarrow M$ (the covering corresponding to this subgroup) 
  \cite[Corollary~V~7.5 and Theorem~V~10.2]{massey}.

  Hence $M_d$ is also an oriented closed connected surface. As one can
  see by using a \cw-decomposition (or by calculating the volume if
  the manifolds are hyperbolic), 
  \[ \chi(M_d) = d \cdot \chi(M).\]
  Thus we get for the genus
  \begin{align*}
    g(M_d) & = \frac{- \chi(M_d) + 2}{2} = \frac{- d \cdot \chi(M) +2}{2}\\
           & = d \cdot g -d +1,
  \end{align*}
  i.e., $M_d$ and $F_{d\cdot g -d +1}$ are homotopy equivalent
  \cite[Theorem~I~8.2]{massey}. 
  Applying Corollary~\ref{cov} and Lemma~\ref{fg2} results in
  \begin{align*}
    \sv{M} & =    \frac{1}{d} \cdot \sv{M_d} 
             =    \frac{1}{d} \cdot \sv[big]{F_{g(M_d)}} \\ 
           & \leq \frac{1}{d} \cdot 
                  \bigl( 4g(M_d) -2\bigr) \\
           & = 4g -4 + \frac{2}{d}.
  \end{align*}
  This inequality holds for all~$d \in \N_{\geq2}$. Therefore it follows
  that
  \[ \sv{M} \leq 4g-4. \qedhere\]
\end{bew}

The estimate of Theorem~\ref{fg} is in fact an equality
(Example~\ref{exfg}).

Additionally, one can use Corollary~\ref{cov} to prove a toy version
of the proportionality principle (Theorem~\ref{propprinc}):

\begin{lem}
  \tindex{proportionality principle!toy version of}
  Let $M$ and $N$ be two oriented closed connected Riemannian
  manifolds. Suppose there is an oriented closed connected Riemannian
  manifold~$U$ admitting locally isometric
  covering maps $U \longrightarrow M$
  and $U \longrightarrow N$, both with finitely many sheets. 
  Then
  \[ \frac{\sv M}{\vol (M)} = \frac{\sv N}{\vol (N)}. \]
\end{lem}
\begin{bew}
  In this situation, both the (Riemannian) volume and the simplicial
  volume are multiplicative with respect to these coverings
  (Corollary~\ref{cov} and \cite[Exercise~14.5]{leesmooth}).
\end{bew}

A proof of the general case requires much more effort. It will be
given in Chapter~\ref{chpropprinc}, based on the results about measure
homology in Chapters \ref{chmh}~and~\ref{chisomisom}.


\subsection{Other volumes}\label{ovol}

There is a great variety of connections between the simplicial volume
and other volumes arising in differential geometry. We have already
seen a distant relation between the simplicial volume and the
Riemannian volume in Lemma~\ref{svvol}. Gromov used the following fact
about hyperbolic manifolds in his proof of Mostow rigidity:

\begin{satz}\label{hypp}%
  \tindex{simplicial volume!of hyperbolic manifolds}%
  \nindex{vn}{$v_n$}{maximal volume of an ideal $n$-simplex in
  hyperbolic space}
  \hypproptmp
\end{satz}
\begin{bew}
  The estimate $\frac{\sv M}{\vol M} \geq \frac 1{v_n}$ can be shown by using the
  so-called straightening of simplices \cite[Corollary~6.1.7]{thurston}, 
  \cite[Lemma~3 in~\S11.4]{ratcliffe}.
  It is possible to use the smearing technique (which will be
  explained in Section~\ref{smearing}) to prove the
  reverse inequality \cite[Theorem~6.2]{thurston}, \cite[page~235]{gromov}. 
  But there are also more direct proofs \cite[Theorem~11.4.3]{ratcliffe}. 
\end{bew}

In particular, there are oriented closed connected manifolds whose
simplicial volume is not zero since $v_n < \infty$ \cite[Lemma~C.2.3]{bp}.

\begin{bsp}\label{exfg}
  \tindex{simplicial volume!of surfaces}
  Let $M$ be an oriented closed connected surface of genus~$g \in
  \N_{\geq 2}$.
  In view of the homotopy invariance of the simplicial volume we may
  assume that $M$ is hyperbolic (since there is a hyperbolic model of
  the surface with genus~$g$ 
  -- a good description of the hyperbolic structure on
  surfaces is given in the book of Benedetti and~Petronio 
  \cite[Section~B.3]{bp}). 
  Hence the Gau\ss-Bonnet formula 
  yields
  \[ \vol(M) = 2\pi \cdot \chi(M) = 2\pi \cdot (2g -2).\]  
  Moreover, $v_2 = \pi$ \cite[Lemma~C.2.3]{bp}. 
  Thus Theorem~\ref{hypp} implies
  \[ \sv{M} = 4g- 4. \]
  In particular, the simplicial volume can grow arbitrarily large and
  it is no bordism invariant. 
\end{bsp}

Based on straightening, Inoue and Yano found the following
generalisation of Theorem~\ref{hypp} \cite{yano}: 

\begin{satz}
  For each $n\in \N$ there is a constant $C_n \in \R_{>0}$ such that
  the following property holds: 
  Let $\delta \in \R_{>0}$ and let $M$ be an $n$-dimensional oriented
  closed connected Riemannian manifold whose sectional curvature is
  bounded from above by~$- \delta.$ Then
  \[ \sv{M} \geq\ C_n \cdot \delta^{n/2} \cdot \vol(M).\]
\end{satz}

The proportionality principle can also be seen as a generalisation of
Theorem~\ref{hypp}. Chapters~\ref{chboco}, \ref{chmh} and~\ref{chisomisom} 
provide the tools necessary for the
proof (based on Thurston's ideas) of this theorem in Chapter~\ref{chpropprinc}.

\tindex{proportionality principle!of simplicial volume}%
\tindex{simplicial volume!proportionality principle}%
\svpropprinc

Gromov showed that
the simplicial volume also can be seen as a topological obstruction
for the existence of certain Riemannian metrics
\cite[Section~0.5]{gromov}:

\begin{defi}%
  \dindex{minimal volume}%
  \dindex{volume!minimal}%
  \nindex{minvol}{$\minvol(M)$}{minimal volume}%
  \nindex{sec}{$\sec_g(M)$}{sectional curvature of~$M$ with respect to the Riemannian
  metric~$g$}%
  Let $M$ be a smooth manifold. Then its \defin{minimal volume},
  $\minvol(M)$, is the infimum over all volumes $\vol_g(M)$, where $g$
  is a complete Riemannian metric on~$M$ such that the sectional
  curvature satisfies $|\sect_g(M)| \leq 1$. 
\end{defi}

\begin{bsp}
  Using the Gau\ss -Bonnet formula, one can show that the minimal
  volume of the oriented closed connected surface of genus~$g \in
  \N_{\geq 2}$ is~$2\pi \cdot | 2 - 2g|$ \cite[page~213]{gromov}. 

  Scaling Riemannian metrics yields $\minvol(M) = 0$ for all flat
  manifolds. 

  A rather counterintuitive result is that $\minvol(S^3) = 0$, as was
  found out by Berger \cite[page~215]{gromov}. 
\end{bsp}

According to Gromov, the minimal volume can be estimated from below 
as follows \cite[Section~0.5]{gromov}: 

\begin{satz}\label{minvolsv}
  For all $n$-dimensional oriented closed connected smooth
  manifolds~$M$, the simplicial volume is bounded from above by the
  minimal volume via
  \[ \sv M \leq (n-1)^n \cdot n! \cdot \minvol(M). \]
\end{satz}

\begin{qedkoro}%
  \tindex{simplicial volume!of flat manifolds}
  In particular, if $M$ is an oriented closed connected flat
  manifold, then $\sv M = 0$. 
\end{qedkoro}

We will prove this corollary as an application of the proportionality
principle in Section~\ref{ppapp}, without making use of Theorem~\ref{minvolsv}.


\subsection{A dual point of view}

Gromov also discovered that bounded cohomology can be used to
investigate the simplicial volume \cite[Chapter~1]{gromov}.
We will sketch this approach in Chapter~\ref{chboco}.  
One of the main results is that the simplicial volume can be
computed in terms of bounded cohomology of groups
(Corollary~\ref{svbg}). This has the following rather striking
consequences: 

\begin{satz}
  \begin{enumerate}
    \item If $M$ is an oriented closed connected manifold with
          amenable fundamental group, then~$\sv M = 0$.
    \item If $f \colon M \longrightarrow N$ is a map of degree~$1$ between
          oriented closed connected manifolds of the same dimension 
          such that $\ker \pi_1(f)$ is amenable, then 
          \[ \sv M = \sv N. \] 
  \end{enumerate}
\end{satz}

The class of amenable groups contains, for example, all Abelian groups
and all finite groups,  
but it does not contain the free product~$\Z * \Z$. In particular, we
see again that $\sv[norm]{S^n} = 0$ for all $n \in \N_{>0}$ and
that $\sv[norm]{S^1 \times S^1} = 0$. More details will be given in
Sections \ref{amenability}~and~\ref{apptosv}. 

Other consequences of the duality between $\lone{\,\cdot\,}$ and
bounded cohomology are (Theorem~\ref{prod} and Theorem~\ref{consum}):

\begin{satz}\label{svprcs}
  \begin{enumerate}
   \item For each $n \in \N_{>0}$ there is a constant $c(n) \in \R_{>0}$ such
         that the following holds: if $M$ and $N$ are two oriented closed
         connected manifolds, then
         \[      \sv N \cdot \sv M 
            \leq \sv{N \times M} 
            \leq c(\dim N + \dim M) \cdot \sv N \cdot \sv M. \]
   \item If $M$ and $N$ are two oriented closed connected manifolds of
         the same dimension~$n \geq 3$, then
         \[ \sv{M \consum N} = \sv M + \sv N. \]
  \end{enumerate}
\end{satz}

In particular, for all oriented closed connected manifolds~$M$, the
simplicial volume of the product $S^1 \times M$ is zero. This is 
generalised in the following way \cite{yanos1}:

\begin{satz}\label{s1sv}%
  \tindex{simplicial volume!S1actions@$S^1$-actions and}
  Suppose $M$ is an oriented closed connected manifold admitting a
  non-trivial $S^1$-action. Then $\sv M = 0$.
\end{satz}

\subsection{Which simplicial volumes are possible?}
\tindex{simplicial volume!realisability of}

In this paragraph we want to analyse which values occur as the
simplicial volume of some oriented closed connected manifold. 

For each $v \in \R_{>0}$ there is an oriented closed connected
Riemannian manifold~$M$ such that $\vol(M) = v$. A very clumsy
estimate, based on the following theorem, shows that this cannot be
true for the simplicial volume:

\begin{satz}
  There are only countably many homotopy types  
  \cw-complexes consisting of finitely many cells.
\end{satz}
\begin{bew}
  We prove by induction (on $k \in \N$) that there are only countably
  many homotopy types of \cw-complexes having at most $k$ cells. 

  If $k = 0$, there is nothing to show. For the induction step let
  $k\in \N$, and let $Y$ be a \cw-complex consisting of $k+1$ cells. 
  Then there is a \cw-complex~$X$ with $k$
  cells and an attaching map $f \colon S^n \longrightarrow X$ for some
  $n \in \N$ such that
  \[ Y \cong X \cup_f D^{n+1}. \]
  Here the homotopy type of~$Y$ depends only on the homotopy class
  of~$f$ and the homotopy type of~$X$ \cite[Corollary~IV~2.4]{lundell}. 

  By induction, there are only countably many homotopy types of
  such~$X$. 
  Hence we have to count the number of homotopy classes of possible
  attaching maps.
  Let $s \in S^n$ and $x \in X$ be two base points. If $x$ is in the
  same connected component as~$f(S^n)$, then $f$ is homotopic to a
  pointed map~$S^n \longrightarrow X$ because~$\{s\}
  \hookrightarrow S^n$ is a cofibration. But there are only countably
  many homotopy classes of pointed maps $S^n
  \longrightarrow X$ because $\pi_n(X,x)$ is countable
  \cite[Theorem~IV~6.1]{lundell}. Moreover, the finite
  \cw-complex~$X$ can only have finitely many connected
  components, and there are, of course, only countably many possible
  dimensions~$n \in \N$.

  Hence there can be only countably many homotopy types of
  \cw-complexes with $k+1$ cells.
\end{bew}
  

\begin{koro}
  The set
  \[ \bigl\{ \sv M
     \bigm|  \text{$M$ is an oriented closed connected manifold}
     \bigr\} 
     \subset \R_{\geq 0}
  \]
  is countable. In particular, not every positive real number can be
  realised as simplicial volume of some oriented closed connected manifold.
\end{koro}
\begin{bew}
  Each closed (oriented, connected) topological manifold has the
  homotopy type of a finite \cw-complex \cite[page~744]{kirby}. By
  the previous theorem, there are only countably many homotopy types
  of closed (oriented, connected) manifolds. Now Corollary~\ref{svhi}
  completes the proof.
\end{bew}

This corollary shows that the simplicial volume has a much more subtle
behaviour than the Riemannian volume: the simplicial volume cannot be
affected by ``scaling.'' 

The corollary gives, of course, a very unsatisfying and rough answer. 
It would be
interesting to know whether the simplicial volumes lie densely 
in~$\R_{>0}$. 
It would be sufficient to know that there is
a sequence $(M_n)_{n \in \N}$ of
oriented closed connected manifolds (of dimension at least 3) with
non-zero simplicial volume such that
\[ \lim_{n \rightarrow \infty} \sv{M_n} = 0. \]
Then one could use Gromov's formula for connected sums
(Theorem~\ref{svprcs})  
to see that the simplicial volumes lie dense in~$\R_{>0}$.

Example~\ref{exfg} shows at least that the simplicial volume can 
become arbitrarily large.

The question of realisability has already been posed by Gromov
\cite[page~222]{gromov}. He suspects that the simplicial volumes might
form a closed well-ordered (with the standard ordering on the real
line) non-discrete subset of~$\R$. This speculation is based on a
result of Thurston on hyperbolic manifolds of dimension~3
\cite{gromovhyp}, \cite{thurston}. 
However, until today there do not seem to
exist any detailed answers to all these questions. 

\section{Simplicial volume and $L^2$-invariants}\label{l2sec}%
\tindex{simplicial volume!L2invariants@$L^2$-invariants and}%
\tindex{L2invariants@$L^2$-invariants}

As mentioned in the introduction, there seems to be a deep 
connection between the simplicial volume and $L^2$-invariants, such as
$L^2$-Betti numbers and $L^2$-torsion. 
More concretely, Gromov formulated the following conjecture
\cite[Section 8A]{gromovasym}:

\Gromovconjecture

L\"uck extended this conjecture as follows \cite[Conjecture~3.2]{lueck2}:

\emph{Moreover, $\ucov M$ is of determinant class and 
$\ltt[big]{\ucov M; {\mathcal N}(\pi_1(M))} = 0$.}

Since aspherical spaces are the central objects of concern, we recall
the definition:

\begin{defi}%
  \dindex{aspherical}
  A space is called \defin{aspherical} if its universal covering is
  contractible. (For \cw-complexes this is the same as to require that
  all higher homotopy groups vanish).
\end{defi}

The $L^2$-Betti numbers~$b_*^{(2)}$ and $L^2$-torsion~$\rho^{(2)}$ can be 
defined (in the setting of von Neumann algebras)
similarly to their classic pendants \cite[Chapter~6]{lueck}. 

Until now there does not seem to exist a strategy to prove (or
disprove) the above conjecture.  However, there are many facts
witnessing the probable truth of this conjecture. In the following
paragraphs we will present some of this evidence. A more extensive
list can be found in L\"uck's book~\cite[Chapter~14]{lueck}. For
simplicity we refer to the proofs in this book instead to the
original papers. 


\subsection{The zeroth $L^2$-Betti number}

A rather easy argument shows that the above conjecture holds for
the zeroth $L^2$-Betti number:

\begin{lem}\label{infinitefg}
  Let $M$ be an aspherical connected  closed
  manifold. Then the fundamental group $\pi_1(M)$ is torsion free and
  hence infinite.
\end{lem}
\begin{bew}
  Assume that $\pi_1(M)$ contains a finite cyclic subgroup $G.$ 
  Since $M$ is aspherical and of finite dimension
  $n,$ the cellular chain complex $\cellch[norm]{*}{\ucov{M};\Z}$ is a 
  bound\-ed projective resolution of the trivial $\Z G$-module
  $\Z.$
  This implies that the group homology
  \[ H_*(G;\Z) \cong H_*\bigl(\cellch[norm]{*}{\ucov M;\Z} \otimes_{\Z G} \Z
     \bigr)\]
  vanishes in all degrees greater than~$n$.  

  On the other hand, it is well-known that the group homology of 
  nontrivial finite cyclic groups does not have this property  
  \cite[Theorem 6.2.3]{weibel}. Hence $G$ must be trivial, i.e., 
  $\pi_1(M)$ is torsion free. 
\end{bew}

\begin{qedkoro}
  If $M$ is an aspherical closed connected manifold, then
  \cite[Theorem~6.54(8)]{lueck} 
  \[ \ltb[big]{0}{\ucov{M};{\mathcal N}(\pi_1(M))} = \frac{1}{| \pi_1(M) |} = 0. 
     \qedhere\]
\end{qedkoro}

\begin{bsp}
  The condition that the manifold be aspherical in Gromov's conjecture
  is necessary: 
  The manifold $S^2$ is closed and orientable,
  but not aspherical because $\pi_2(S^2) \cong \Z.$ By Corollary 
  \ref{svselfmap}
  we have $\sv[norm]{S^2} = 0.$ On the other hand
  \cite[Theorem~6.54(8)]{lueck}, 
  \[ \ltb[big]{0}{\ucov{S^2};{\mathcal N}(\pi_1(S^2))} 
    = \frac{1}{|\pi_1(S^2)|} = 1 \neq 0.\qedhere\]
\end{bsp}

\subsection{Amenability}

If $M$ is on oriented closed connected (not necessarily aspherical)
manifold with amenable fundamental group, then $\sv M = 0$
(Corollary~\ref{svbg}). The corresponding result in the $L^2$-world
is:

\begin{satz}
  Let $M$ be an oriented closed connected aspherical manifold with
  amenable fundamental group. Then all $L^2$-Betti numbers of~$M$ are
  zero.  
\end{satz}
\begin{bew}
  By Lemma~\ref{infinitefg}, the fundamental group of~$M$ must be
  infinite. But then it can be shown that all $L^2$-Betti numbers of~$M$
  must be zero \cite[Corollary~6.75]{lueck}.
\end{bew}


\subsection{Proportionality}

One of the most remarkable similarities between the simplicial volume
and $L^2$-invariants is the proportionality principle. 
It is valid for the simplicial volume (Theorem~\ref{propprinc})
and for 
$L^2$-Betti numbers (and $L^2$-torsion) \cite[Theorem~3.183]{lueck}:

\begin{satz}[Proportionality Principle of $L^2$-Invariants]\label{l2pp}%
  \tindex{proportionality principle!of $L^2$-invariants}
  \ltproptmp
\end{satz}

\subsection{Multiplicativity}%
\tindex{multiplicativity}

As we have seen in Section \ref{svpropsec}, the simplicial volume is
multiplicative with respect to the mapping degree. Therefore Gromov's
conjecture predicts that if $M$ is an aspherical oriented closed connected
manifold admitting a self-map of degree at least two, then all
$L^2$-Betti numbers of $M$ vanish. This statement is known to be true
if this self-map is a covering~\cite[Example~1.37]{lueck}.

The general case is still open, but there is at least a partial result:

\begin{defi}%
  \dindex{Hopfian}
  A group $G$ is said to be \defin{Hopfian} if every epimorphism $G
  \longrightarrow G$ is an isomorphism.
\end{defi}

Examples for Hopfian groups are finite groups, the integers, and, 
more generally, residually finite groups. The group $\ell^{\infty}(\N,\R)$
is not Hopfian as can be seen by considering the shift operator.

\begin{satz}
  Let $M$ be an aspherical oriented closed connected manifold. If
  there is a continuous map $f \colon M \longrightarrow M$ such that
  $|\deg(f)| \geq 2$ and
  if all subgroups of~$\pi_1(M)$ of finite index are Hopfian, then
  \[ \fa{p \in\N} \ltb[big]{p}{\ucov M;{\mathcal N}(\pi_1(M))} = 0.\]
\end{satz}

A proof is given in~\cite[Theorem~14.40]{lueck}. (There only all
\emph{normal} subgroups of~$\pi_1(M)$ are required to be Hopfian --
but this is a misprint.)




\subsection{Non-trivial $S^1$-operations}

Furthermore, the simplicial volume and the $L^2$-Betti numbers behave
similarly under $S^1$-operations \cite[Corollary~1.43]{lueck}:

\begin{satz}
  Let $M$ be an aspherical closed smooth manifold with non-trivial
  $S^1$-action. Then 
  \[\fa{p \in\N} \ltb[big]{p}{\ucov M; {\mathcal N}(\pi_1(M))} = 0.\]
\end{satz}

The corresponding
statement about the simplicial volume is~Theorem~\ref{s1sv}.


\subsection{\protect\reflectbox{Gromov's conjecture}}

Looking at the conjecture and the known evidence one could also
suspect that the converse is true: If $M$ is an aspherical oriented
closed connected manifold whose $L^2$-Betti numbers vanish, then
$\sv{M} = 0.$

But this statement is false: If $M$ is an oriented closed connected
hyperbolic manifold of odd dimension, then 
\[ \ltb[big]{p}{\ucov M; {\mathcal N}(\pi_1(M))} = 0 \]
for all $p\in \N$~\cite[Theorem~1.54]{lueck}. On the other hand, 
$\sv{M} \neq 0$ by Theorem~\ref{hypp}.



% -------------------------------------------------------------------
\section{Generalisations}\label{svgen}%
\tindex{simplicial volume!generalisations}

In the following, some possible generalisations of the simplicial
volume are described. In the first part, we consider the use of
coefficients other than~$\R$. Later we sketch extensions of the simplicial 
volume to non-compact, non-connected, or non-orientable manifolds
possibly with non-empty boundary, and also to Poincar\'e complexes. 
In the last section we consider functorial seminorms in the sense of
Gromov. 

\subsection{Unreal coefficients}

Instead of real coefficients we could also use the singular chain
complex with integral, complex or rational coefficients to define ``simplicial
volumes'' $\sv M _{\Z}$, $\sv M _{\C}$, and $\sv M _{\Q}$. 

\begin{defi}
  Let $R \in \{ \Z, \C, \Q\}$ and let $M$ be an oriented closed
  connected manifold of dimension~$n$. We write
  $ \fcl M _R \in \hsingc n M R \cong R$ for the image of the fundamental
  class~$\fcl M$ under the change of coefficients homomorphism. 
  \nindex{$v}{$\fcl M_R$}{fundamental class with $R$-coefficients}

  As in the case of singular homology with real coefficients, we can
  define an ``$R$-linear norm'' on~$\csingc n M R$ via
  \[ \fsum_{\sigma \in \ssing n M} a_\sigma \cdot \sigma 
     \longmapsto \fsum_{\sigma \in \ssing n M} |a_\sigma|, \]
  which induces a ``seminorm''~$\lone{\, \cdot \,}^R$ on~$\hsingc n M R$
  \dindex{seminorm}%
  \nindex{$j}{$\lone{\,\cdot\,}^R$}{``seminorm''}
  by taking the infimum over all representatives. That is, we define 
  \[ \lone \alpha ^R := \inf \bigl\{ \lone a ^R 
                             \bigm| \text{$a \in \csingc n M R$, 
                                          $\bou a = 0$, 
                                          $[a] = \alpha \in \hsingc n M R$} 
                             \bigr\}\]
  for all~$\alpha \in \hsingc n M R$. 
  In particular, we write
  \[
    \sv M _{\Z} := \lone[big]{\fcl M_{\Z}}^{\Z}, \quad
    \sv M _{\C} := \lone[big]{\fcl M_{\C}}^{\C}, \quad
    \sv M _{\Q} := \lone[big]{\fcl M_{\Q}}^{\Q}. \qedhere
  \]%
  \nindex{$p}{$\sv M _{\Z}$}{}
  \nindex{$n}{$\sv M _{\C}$}{}
  \nindex{$o}{$\sv M _{\Q}$}{}
\end{defi}

Of course, we could also define a notion of volume using other
homology theories based on chain complexes which can be equipped with
some ``norm'' and whose top homology group (of oriented closed
connected manifolds) contains a distinguished class.  

The simplicial volume with integral coefficients obviously never 
vanishes. So it cannot be the same as the usual simplicial
volume. But the integral simplicial volume has an interesting
connection with the Betti numbers of the corresponding
manifold. However, the integral version is not very comfortable to
work with. 


\begin{lem}
 Suppose $M$ is an oriented closed connected $n$-dimensional manifold
 and $\sum_{j = 1}^t a_j \cdot \sigma_j \in \csingc n M \Z$
 represents~$\fcl M_\Z$. Then
 \[ \sum_{k \in \N} b_k(M) \leq t \cdot 2^{n+1},\]
 where $b_k (M) = \rk \hsing k M$ is the $k$-th Betti number of~$M$
 (which is the same as $\dim_\R \hsingc k M \R$ and $\dim_\Q \hsingc k M \Q$).
 In particular, 
 \[ \sum_{k \in \N} b_k(M) \leq 2^{n+1} \cdot \sv M_\Z.\]
\end{lem}

The proof is based on Poincar\'e duality \cite[Example~14.28]{lueck}. 
The following result is in the same direction, but is much harder
to prove \cite[page~220]{gromov}. 

\begin{satz}
  For all $k \in \R_{\geq 1}$ and all $n \in \N$, there is a
  constant~$C(n,k)$ with the following property: If $M$ is an
  $n$-dimensional oriented closed connected Riemannian manifold with
  $-k_1 \leq \sec(M) \leq k_2$ for some $k_1 \geq k_2 >0$, then the
  Betti numbers of~$M$ can be estimated via
  \[ \sum_{k\in \N} b_k(M) \leq C(n, k_1/k_2) \cdot \sv M. \]
\end{satz}

It is not hard to see that the real, complex, and rational simplicial
volume all coincide:

\begin{lem}
  Let $M$ an oriented closed connected manifold of dimension~$n$. 
  Then 
  \[ \sv M _\C = \sv M = \sv M_\Q. \]
\end{lem}
\begin{bew}
  The inclusions $\csingc * M \Q \subset \csingc * M \R \subset
  \csingc * M \C$ yield
  \[ \sv M _\C \leq \sv M \leq \sv M _\Q. \]
  The inequality $\sv M_\C \geq \sv M$ can be shown by considering the real
  part of a representing cycle.
  The inequality $\sv M \geq \sv M_\Q$ can be obtained via an
  approximation of boundaries with real coefficients by 
  boundaries with rational coefficients. 
  In both cases one has to use the fact that the fundamental class
  with rational or complex coefficients comes from a class in homology
  with integral coefficients.
\end{bew}

\subsection{More spaces}

It is also possible to define a simplicial volume in more general
settings \cite[page~216]{gromov}. 
For example, if $M$ is a closed connected manifold which is
not orientable, we can define
\[ \sv{M} := \frac12 \cdot \sv{\overline M}, \]
where $\overline M$ is the orientation double covering of~$M$. 


If $M$ is not connected we can just sum up the simplicial volumes of
the connected components.

If $M$ is an oriented connected manifold without boundary which is not
compact, we can use the fundamental class of~$M$ in locally finite
homology to define a (possibly infinite) simplicial volume. 

If $M$ is a manifold with boundary, then we can define the simplicial
volume as the seminorm of the relative fundamental class~$[M,\partial M]$. 

Since the definition of the simplicial volume only makes use of the
homological properties of manifolds, it is easy to find a definition
of simplicial volume for homology manifolds (with respect to real
coefficients) or Poincar\'e complexes, which are some kind of
``homotopy manifolds.''

All these extensions preserve basic properties, such as homotopy
invariance, multiplicativity etc. 

\subsection{Functorial seminorms}%
\tindex{functoriality}%
\tindex{seminorm}

Instead of the seminorm in singular homology with real coefficients
induced by the $\ell^1$-norm, one could also use other seminorms. To
ensure that the resulting invariant has useful properties, Gromov
introduced the following notion \cite[5.34]{gromovb}:

\begin{defi}
  Suppose for each topological space~$X$ a seminorm~$\| \, \cdot \,\|$
  is given on~$\hsingc * X \R$. Such a seminorm is called
  \defin{functorial}%
  \dindex{functorial seminorm} 
  if for each continuous map~$f \colon X
  \longrightarrow Y$ and all $\alpha \in \hsingc * X \R$
  \[ \| \hsingc * f \R (\alpha)   \| \leq \| \alpha \|. \qedhere\] 
\end{defi}

By Lemma~\ref{lonenfunc} the seminorm $\lone{\, \cdot\,}$ on homology with real
coefficients induced by the $\ell^1$-norm on the singular chain
complex is functorial. 
Other examples are given in Gromov's book \cite[pages~302$-$316]{gromovb}.

The proof of Corollary~\ref{mult} of course shows that each functorial
seminorm is submultiplicative with respect to the degree of maps. 

\begin{defi}
  If $\|\,\cdot\,\|'$ is a functorial seminorm, one can define an
  analogue of the simplicial volume as the value of~$\|\,\cdot\,\|'$ on
  the fundamental class.
\end{defi}

Then the same argument as in Corollary~\ref{svselfmap} shows that $\|S^n\|'$
must be zero for all $n \in \N_{> 0}$.

Using bounded cohomology, we will see in Corollary~\ref{bcafg} that the
seminorm induced by the $\ell^1$-norm vanishes on simply connected
spaces. Gromov conjectured that this phenomenon occurs for all
functorial seminorms \cite[Remark in~5.35]{gromovb}: 

\begin{conj}
  Any functorial seminorm on singular homology with real coefficients
  vanishes for simply connected spaces. 
\end{conj}

Until today, neither a proof nor counterexamples to this amazing
conjecture are known. 



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