The fundamental group of $G$-manifolds
Hui Li
Let $M$ be a connected smooth $G$-manifold, where $G$ is a connected compact Lie group. In this paper, we first study the relation between $\pi_1\left(M\right)$ and $\pi_1\left(M/G\right)$. Then we particularly focus on the case when $M$ is a connected Hamiltonian $G$-manifold with an equivariant moment map $\phi$. In \cite{L2}, for {\em compact} $M$, we proved that $\pi_1\left(M\right)\cong \pi_1\left(M/G\right)\cong\pi_1\left(M_a\right)$ for all $a\in \mbox{image}(\phi)$, where $M_a$ is the symplectic quotient at $a$. We generalize and extend these results to Hamiltonian $G$-manifolds with {\em proper} moment maps. Our main results are Proposition~\ref{M=M/G}, Theorems~\ref{thm2} and ~\ref{thm3}.
0909.3179.tex

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\begin{document}
\title[The fundamental group of $G$-manifolds]{The fundamental group of $G$-manifolds}
\author{Hui Li} 
\address{Institut de Mathematiques de  Bourgogne\\
         9 Av. Alain Savary, 21078, Dijon, France.}
\email{hui.li@u-bourgogne.fr}


\thanks{2000 MSC. Primary :  53D05, 53D20; Secondary : 55Q05, 57R19.}
\keywords{Symplectic manifold, fundamental group, Hamiltonian group
action, moment map, symplectic quotient.}
\begin{abstract}
 Let $M$ be a connected smooth $G$-manifold, where $G$ is  a connected compact Lie group.
 In this paper, we first study the relation between $\pi_1\left(M\right)$
 and $\pi_1\left(M/G\right)$. Then we particularly focus on the case when
 $M$ is a connected Hamiltonian $G$-manifold with an equivariant moment map $\phi$.
 In \cite{L2}, for {\em compact} $M$, we proved that  $\pi_1\left(M\right)\cong \pi_1\left(M/G\right)\cong\pi_1\left(M_a\right)$
 for all $a\in \mbox{image}(\phi)$, where $M_a$ is the symplectic quotient at
$a$. We generalize and extend these results to Hamiltonian
$G$-manifolds with {\em proper} moment maps.
Our main results are Proposition~\ref{M=M/G}, Theorems~\ref{thm2} and ~\ref{thm3}.
\end{abstract}
 \maketitle
 \section{Introduction}


  Let $M$ be a smooth manifold. Let a connected compact Lie group $G$ act
  on $M$. We call $M$ a {\em $G$-manifold}.

  Let $(M, \omega)$ be a symplectic manifold. Assume that a connected compact Lie group $G$
  acts on $M$  with  moment map  $\phi: M\rightarrow \mathfrak{g}^*$,
   where $\mathfrak{g}^*$ is the dual Lie algebra of $G$. In this case, we call $(M, \omega)$ a {\em Hamiltonian $G$-manifold}.
 We will always assume that $\phi$ is equivariant with respect to the $G$ action, where $G$ acts on $\mathfrak{g}^*$ by the coadjoint action. Take a moment map value
 $a$ in $\mathfrak{g}^*$, the space ${\bf M_{a}}=\phi^{-1}(G\cdot a)/G$ is called the {\bf symplectic quotient}
  or the {\bf reduced space} at the coadjoint orbit $G\cdot a$. If $G_a$ is the stabilizer group of $a$ under
  the coadjoint action, by equivariance of the moment map, ${\bf M_a}=\phi^{-1}(a)/G_a=\phi^{-1}(G\cdot a)/G$. We will use
 the two notations interchangeably.


 In this paper,  fundamental groups always mean the fundamental
 groups of topological spaces, even when we do not explicitly say so.\\


 For compact Hamiltonian $G$-manifold $M$, in \cite{L2}, we had the following results.

  \begin{theorem}\label{thm1}
     Let $(M, \omega)$ be a connected compact Hamiltonian $G$-manifold
  with equivariant moment map $\phi$, where $G$ is a connected compact
   Lie group. Then, as fundamental groups
   of topological spaces, $\pi_1\left(M\right)\cong \pi_1\left(M/G\right)\cong\pi_1\left(M_a\right)$ for all
  $a\in \mbox{im}(\phi)$.
   \end{theorem}

   If $M$ is not compact, the results in Theorem~\ref{thm1} may not be true as the following
   easy example shows.
\begin{example}\label{ex}
 Consider
  $M=S^1\times\R$, and let $S^1$ rotate the first factor, then  the
  moment map is the projection to the second factor $\R$. We have that
  $\pi_1\left(M/S^1\right)\cong\pi_1\left(M_a\right)\ncong\pi_1\left(M\right)$
 for all $a\in \mbox{im}(\phi)$.
\end{example}



   The proof of Theorem~\ref{thm1} goes as follows (see \cite{L2} and \cite{L1}): we prove that
   $\pi_1\left(M\right)\cong\pi_1\left(M_a\right)$ for {\em some value $a$} which is a {\em boundary vertex}
   of the moment polytope. In fact, for this special value $a$,
  the whole set $\phi^{-1}(a)$ is fixed by $G_a\supseteq T$, where $G_a$ is the stabilizer group of $a$ under the coadjoint action,
 and $T$ is a maximal torus of $G$. Therefore, $\phi^{-1}(a)\approx \phi^{-1}(a)/G_a=M_a$. So
\begin{equation}\label{a}
  \pi_1\left(M\right)\cong\pi_1\left(\phi^{-1}(a)\right)\cong\pi_1\left(M_a\right).
\end{equation}
  Then we
   prove that the $\pi_1(M_a)$'s are isomorphic for all value $a$'s.
   Finally, we prove that $\pi_1\left(M/G\right)\cong \pi_1\left(M_a\right)$ for a
   particular value $a$.

  \begin{lemma}\label{fix}
 Let $(M, \omega)$ be a connected Hamiltonian $G$-manifold, where $G$
   is a connected compact Lie group. Let $T$ be a maximal torus of $G$.
 If $M$ is {\em compact}, then there exists a connected subgroup $H$ of $G$ such that
 $H\supseteq T$ and that $H$  has a fixed point.
 \end{lemma}
 \begin{proof}
 Let $\phi$ be the moment map.
 If $G=T$ is abelian, then $\phi(M)$ is a convex polytope for compact $M$. The inverse image of each boundary
 vertex of the polytope is a fixed component of $T$. If $G$ is nonabelian, let $a$ be a furthest value in the image of $\phi$.
 In Lemma 6.14  of \cite{L2}, although we did not state it, we proved that $\phi^{-1}(a)$ is fixed by $G_a\supseteq T$, where $G_a$
 is the stabilizer group of $a$ under the coadjoint action.
 \end{proof}


   If $M$ is not compact, as we saw in  Example~\ref{ex}, there may not be isotropy group which contains
 a maximal torus of $G$; and, for this example, $\pi_1\left(M\right)\ncong\pi_1\left(M/G\right)$.
  While for compact $M$, Lemma~\ref{fix} holds, and Theorem~\ref{thm1} says that $\pi_1\left(M\right)\cong\pi_1\left(M/G\right)$. \\


  Now, we only assume that $M$ is a connected
 $G$-manifold with moment map $\phi$. The moment map $\phi$ is {\bf proper}
  if the inverse image under $\phi$ of each compact set is compact.

 For a noncompact Hamiltonian $G$-manifold $M$ with proper moment map $\phi$, $\phi(M)\cap\mathfrak t^*_+$
 is a convex polyhedral set (Theorem~\ref{convexity}),  where $\mathfrak t^*_+$
 is a closed positive Weyl chamber. But, $\phi(M)$ may not be
 compact and hence there may not be boundary vertex on $\phi(M)\cap\mathfrak t^*_+$. We will take a different approach from that of compact $M$ to study $\pi_1(M)$. First, we study $\pi_1(M)$ for general $G$-manifolds.




 Let  a connected compact Lie group $G$ act on a connected smooth manifold $M$.
 Let $H$ be a subgroup of $G$ which is the stabilizer of some point in $M$. Let $(H)$ be the
subgroups of $G$ conjugate to $H$. Denote $M_{(H)}=\{ x\in M\, |\,
G_x\in (H)\}$. The set $M_{(H)}$ is a submanifold of $M$, and we
call it the {\bf $(H)$-orbit type} or the {\bf $(H)$-isotropy type}.
There is a {\em connected open dense} orbit type in $M$, which is
called the {\bf principal orbit type}. We first have the following
result.

\begin{proposition}\label{M=M/G}
 Let $G$ act on a connected  smooth manifold $M$, where $G$ is a connected compact Lie group.
 Assume that  each nonprincipal orbit type
 has codimension at least $2$, and that there exists an  orbit which is simply connected.
 Then
$$\pi_1(M)\cong\pi_1\left(M/G\right).$$
 \end{proposition}



 If we assume that big enough isotropy group $H\supseteq T$ exists for the $G$ action, then
 there exists a simply connected orbit (see Lemma~\ref{G/T}). Now, let us consider Hamiltonian $G$-manifolds.
 Our next main result is Theorem~\ref{thm2}, where (A) holds without assuming the moment map is proper, and 
 we assume the moment map is proper in (B).




 \begin{theorem}\label{thm2}
   Let $(M, \omega)$ be a connected Hamiltonian $G$-manifold with  equivariant moment map $\phi$, where $G$
   is a connected compact Lie group. Let $T$ be a maximal torus of $G$.
Assume that there is a connected subgroup $H$ of $G$ such that $H\supseteq T$
and that $H$ has a fixed point. Then
 \begin{itemize}
  \item [(A)]  $\pi_1\left(M\right)\cong\pi_1\left(M/G\right)\cong\pi_1\left(M/T\right)\cong\pi_1\left(M/H\right).$
  \item [(B)]
  Let $x$ be the $H$ fixed point. If $\phi$ is proper, and if $\phi(x)=a$, then
 $$\pi_1\left(\phi^{-1}(G\cdot
 a)\right)\cong\pi_1\left(\phi^{-1}(a)\right)\cong\pi_1\left(M_a\right).$$
\end{itemize}
  \end{theorem}

\begin{remark}
 In Theorem~\ref{thm2}, if there is a subgroup $H$ such that
 $H\supseteq T$ has a fixed point, then  the identity
 component of $H$ has a fixed point. So we can always take connected
 $H$. In fact, all the isomorphisms except $\pi_1\left(M\right)\cong\pi_1\left(M/H\right)$ do not need
$H$ to be connected. We take a connected group to make $M$ to be a Hamiltonian $H$-manifold for
 the connected group $H$.
\end{remark}


 Next,  we observe that, the fact $\pi_1\left(M/G\right)\cong \pi_1\left(M_a\right)$
   for all $a\in\mbox{im}(\phi)$ always holds for Hamiltonian $G$-manifolds
   with proper equivariant moment maps. This is a straightforward generalization
 of the compact case.  See Section~\ref{sectionM/G} for the proof. We observe that this result does not depend on whether the action
  has big enough isotropy groups. In our next main result
  Theorem~\ref{thm3} below, we first state this generalization in (A).
 In (B), we made a step further in generalizing (A). For noncompact $M$,
 we observed that there may not be isotropy groups which contain a maximal torus of $G$.
 If we take $\Gh$ to be a connected subgroup of $G$
 such that up to  finitely many elements,  every isotropy group is a subgroup of $\Gh$, we
 can express $\pi_1\left(M/\Gh\right)$
in terms of $\pi_1$  of the ``partial'' reductions of the $G$-moment map by $\Gh$.
 \begin{theorem}\label{thm3}
      Let $(M, \omega)$ be a connected Hamiltonian $G$-manifold
      with a proper equivariant moment map $\phi$, where $G$
   is a connected compact Lie group.
 \begin{itemize}
\item [(A)] $\pi_1\left(M/G\right)\cong\pi_1\left(M_a\right)$ for all $a\in \mbox{im}(\phi).$
\item [(B)] More generally, let $\Gh$ be any connected subgroup of $G$ such that up to finitely many elements, every isotropy group is a subgroup of $\Gh$. Then,
$\pi_1\left(M/\Gh\right)\cong\pi_1\left(\phi^{-1}(G\cdot a)/\Gh\right)$ for all $a\in\mbox{im}(\phi).$ 
\end{itemize}
\end{theorem}






  A special case is when $\mbox{im} (\phi)$ consists of regular
   values. In this case, all the isotropy groups are finite.
 Taking $\Gh=1$ in Theorem~\ref{thm3},  we obtain
  $\pi_1(M)\cong\pi_1(\phi^{-1}(G\cdot a))$ for
   all $a\in \mbox{im}(\phi)$.  We have the following lemma, where the proof of
  the fact $\pi_1(M)\cong\pi_1\left(\phi^{-1}(a)\right)$
   for all $a\in \mbox{im}(\phi)$ is given below.
   \begin{lemma}\label{M-Gregular}
      Let $(M, \omega)$ be a connected Hamiltonian $G$-manifold
      with a proper equivariant moment map $\phi$, where $G$
   is a connected compact Lie group.
  Assume that $\mbox{image} (\phi)$ consists of regular
   values. Then,
    $$\pi_1(M)\cong\pi_1\left(\phi^{-1}(a)\right)\cong\pi_1\left(\phi^{-1}(G\cdot a)\right) \quad\mbox{for
   all} \quad a\in \mbox{im}(\phi).$$
   \end{lemma}
  \begin{proof} [Proof of $\pi_1(M)\cong\pi_1\left(\phi^{-1}(a)\right)$
   for all $a\in \mbox{im}(\phi)$.]
 If  $\mbox{im}(\phi)$ consists of regular values, then $\phi$ is a proper
 submersion from $M$ onto $\mbox{im}(\phi)\subset\mathfrak{g}^*$.
   By Ehresmann's lemma, $\phi$ gives a fibration from $M$ onto
$\mbox{im}(\phi)$ (which is open in $\mathfrak{g}^*$) with connected
fiber (\cite{K} or \cite{LMTW})
    diffeomorphic to $\phi^{-1}(a)$ for
  {\em any} $a\in \mbox{im}(\phi)$. By the homotopy exact sequence for fibrations, we have
$\pi_1(M)\cong\pi_1\left(\phi^{-1}(a)\right)$.
 \end{proof}
 We have the following corollary of Theorems~\ref{thm2} and \ref{thm3}.
 It states the important contribution to $\pi_1\left(M\right)$ of the level sets which contain fixed points of
 big enough subgroups of $G$. In particular, this is a generalization of the 
fact $\pi_1\left(M\right)\cong\pi_1\left(\mbox{minimum}\right)\cong\pi_1\left(\mbox{maximum}\right)$ for
a compact Hamiltonian $S^1$-manifold $M$, where minimum and maximum are those of the real valued moment map for the circle action (\cite{L1}); and, more generally, it is a generalization of $\pi_1\left(M\right)\cong\pi_1\left(\phi^{-1}(a)\right)$  
as in (\ref{a})  for some boundary vertices of the moment polytope for compact Hamiltonian $G$-manifold $M$. 
 This corollary and (B) of Theorem~\ref{thm2} together
 generalize  (\ref{a}), even though here $H$ may not fix the whole set $\phi^{-1}(a)$.
 

 \begin{corollary}\label{M+level}
 Let $(M, \omega)$ be a connected Hamiltonian $G$-manifold
      with a proper equivariant moment map $\phi$, where $G$
   is a connected compact Lie group. Let $T$ be a maximal torus of $G$.
Assume that there is a connected subgroup $H$ of $G$ such that $H\supseteq T$
and that $H$ has a fixed point $x$, and $\phi(x)=a$. Then
$$\pi_1\left(M\right)\cong\pi_1\left(\phi^{-1}(G\cdot
 a)\right)\cong\pi_1\left(\phi^{-1}(a)\right).$$
\end{corollary}

For $S^1$-manifolds with proper moment maps, in \cite{GSD}, using
Morse theory, Godinho and Sousa-Dias proved that when $S^1$ does not
have fixed points,  $\pi_1(M)\cong\pi_1\left(\phi^{-1}(a)\right)$
for all $a\in\mbox{im}(\phi)$; and, when $S^1$ has a fixed point,
$\pi_1(M)\cong\pi_1\left(M_a\right)$ for all $a\in\mbox{im}(\phi)$.
This result is a partial result of our Theorems~\ref{thm2} and
\ref{thm3} for the case $G=S^1$.











\section{Preliminaries on $G$-manifolds and on Hamiltonian $G$-manifolds}

In this section, we review some theorems on  Hamiltonian
$G$-manifolds which constitute the main tools in our proofs, and we
will explain how we use them. We  review the notion of stratified
spaces which are essential for our  proof of Theorem~\ref{thm3}
since $M/G$ and $M_a$ are in general stratified spaces.

\begin{theorem}\label{convexity}
      (\cite{AT}, \cite{GS0}, \cite{K} and \cite{LMTW})  Let $(M, \omega)$ be a connected  Hamiltonian $G$-manifold with proper moment map $\phi$, where $G$ is a connected compact Lie group. Let $\mathfrak{t}_+^*$
   be a fixed closed positive Weyl chamber of $\mathfrak{g}^*$. Then
 \begin{enumerate}
  \item For each coadjoint orbit $\mathcal{O}$ in the image of $\phi$, $\phi^{-1}(\mathcal{O})$ is
     connected.
 \item The image
   $\phi(M)\cap\mathfrak{t}_+^*$ is a convex polyhedral set.
  \end{enumerate}
     \end{theorem}


  If $G=T$ is abelian, then $\phi(M)$ is a convex polyhedral set; it consists of {\bf faces} with different dimensions.
  A connected open face is called a {\bf chamber}. By quotienting out a
   subtorus which acts trivially, we can assume that the
  action is effective, and hence there is an open face.

 \begin{theorem}\label{retract} (\cite{W} or \cite{Ler})
           Let $(M, \omega)$ be a connected  Hamiltonian $G$-manifold with proper moment map $\phi$,
           where $G$ is a connected compact Lie
           group. Choose a $G$-invariant metric on $M$. Assume that $0\in\mbox{im}(\phi)$.
   Then there exists a $G$-invariant open neighborhood $U\subset\mathfrak g^*$  of
           $0$ such that the gradient flow of the function $-\|\phi\|^2$
           induces a $G$-equivariant deformation retraction
          from  $\phi^{-1}(U)$ to $\phi^{-1}(0)$.
  \end{theorem}

\subsection{Local normal form theorem}

\begin{theorem}\label{form}
        (Local normal form) (\cite{M}, \cite{GS1}) Let $(M, \omega)$ be a symplectic manifold with a Hamiltonian connected compact Lie group $G$ action.
        Let $p\in M$, and assume that the orbit $G\cdot p$ is isotropic. Let
         $H$ be the isotropy subgroup of $p$. Then a neighborhood of the orbit $G\cdot p$ in $M$
         is equivariantly symplectomorphic to
         $G\times_H(\mathfrak{h}^{\circ}\times V)$, where $\mathfrak{h}^{\circ}$ is the annihilator of $\mathfrak{h}=$Lie$(H)$
    in $\mathfrak g^*$ on which $H$ acts by the coadjoint action, and $V$ is
 a complex vector space on which $H$ acts linearly and symplectically.





 The $G$ action on this local model is $g_1\cdot [g, a, v]=[g_1g, a, v]$, and the moment map on this local model is
       $\phi([g, a, v])=Ad^*(g)(\phi(p)+a+\psi(v))$, where $\psi(v)$ is the moment map for the $H$ action on $V$.
       \end{theorem}

 \subsection{Cross section theorem}

  \begin{definition}\label{def 4.1}
   Suppose that a group $G$ acts on a manifold $M$. Given a point $m$ in $M$ with isotropy group $G_m$,
   a submanifold $U_m\subset M$ containing $m$ is a
   {\bf slice at m} if $U_m$ is $G_m$-invariant, $G\cdot U_m$ is a neighborhood of $m$, and the map
 $$G\times_{G_m}U_m\rightarrow G\cdot U_m, \qquad      [g, u]\longmapsto g\cdot u\quad\mbox{is an isomorphism}.$$
   \end{definition}

   For instance, consider the co-adjoint action of $G=SU(2)$ or $SO(3)$ on $\mathbb{R}^3=$Lie$(G)$.
   For $a\in \mathbb{R}^3, a\neq 0$, there is a unique ray $I_a$
   passing through $0$ and $a$; the open ray $I_a^{\circ}=I_a-0$ is a slice at $a$.
   If $a=0$, then a  slice at $0$ is  $\mathbb{R}^3$.


    More generally, consider the co-adjoint action of a connected compact
  Lie group $G$ on $\mathfrak g^*$.
  Fix a closed positive Weyl chamber $\mathfrak t_+^*$, without loss of generality, take
  $a\in\mathfrak t_+^*$. Let $\tau\subset\mathfrak t_+^*$ be the open face of  $\mathfrak t_+^*$ containing
 $a$  and let $G_a$ be the isotropy group of $a$. Since all the points on $\tau$ have the same isotropy group,
 we also use $G_{\tau}$ to denote $G_a$. Then the natural  slice at $a$ is
$U_a=G_a\cdot\{b\in\mathfrak t_+^* \,|\,G_b\subset
G_a\}=G_a\cdot\bigcup_{\tau\subset\bar{\tau'}}\tau'$,
   and it is an open subset of $\mathfrak g_{\tau}^*=\mathfrak g_a^*$.\\



   The following cross section theorem is due to Guillemin and Sternberg.
(Theorem 26.7 in \cite{GS}; for the following version, see Corollary
  2.3.6 in \cite{GLS}.)

    \begin{theorem}\label{cross}
    (cross section). Let $(M, \omega)$ be a symplectic manifold with a moment map $\phi: M\rightarrow \mathfrak{g}^*$ arising from an action of a compact
    connected Lie group $G$. Let $a$ be a point in $\mathfrak{g}^*$ and let $U_a$ be the natural slice at $a$.
     Then the {\bf cross section} $R=\phi^{-1}(U_a)$ is
     a $G_a$-invariant symplectic submanifold of $M$, where $G_a$ is the isotropy group of $a$. Furthermore,
     the restriction $\phi|_R$ is a moment map
     for the action of $G_a$ on $R$.
     \end{theorem}




   The highest dimensional face $\tau^P$ of $\mathfrak{t}_+^*$ which contains values of $\phi$ is called the
   {\bf principal face}. Let $U_{\tau^P}$ be the slice at $\tau^P$. The cross section
 $\phi^{-1}(U_{\tau^P})$ is called the {\bf principal cross section}.
The maximal torus of $G$ acts on the principal cross section (but
not necessarily effectively, see Theorem 3.1 in \cite{LMTW}).\\


  We will use the cross section theorem in the following  two ways:
\begin{itemize}
 \item   If $a\in\mbox{im}(\phi)\cap\tau$ for some face $\tau\subset\mathfrak t^*_+$, then $\tau$ lies
 on the central dual Lie algebra of $G_a$, so the $G_a$-orbits in $\phi^{-1}(a)$ are isotropic orbits;
 then we can use the local normal form theorem  in the cross section $R$ to describe a neighborhood of
 a $G_a$-orbit.
 \item
 Assume that the moment map $\phi$ is proper. Fix a $G$-invariant Riemannian metric on $M$.
 Assume that $a\in\mbox{im}(\phi)\cap\tau$. By Theorem~\ref{retract},
 there exists a $G_a$-invariant neighborhood of  $\phi^{-1}(a)$  in $R$
 which $G_a$-equivariantly deformation retracts to $\phi^{-1}(a)$. By
 equivariance of $\phi$, there exists a $G$-invariant neighborhood
$\mathcal N$ of $\phi^{-1}(G\cdot a)$ in $M$ which $G$-equivariantly
deformation retracts to $\phi^{-1}(G\cdot a)$.
\end{itemize}


\subsection{Stratified space and link of a point}
\ \

Let $G$, a connected compact Lie group, act on a smooth manifold
$M$, the quotient space $M/G$ is a stratified space. If $(M,
\omega)$ is a symplectic manifold and the $G$ action is Hamiltonian,
then the symplectic quotients are stratified spaces (\cite{SL}).

   Let $X$ be a Hausdorff and paracompact topological space and let $\mathcal J$ be a partially ordered set with order relation denoted by $\leq$. A $\mathcal J$-decomposition of $X$ is a locally finite collection of disjoint, locally closed manifolds $S_i\subset X$ (one for each $i\in\mathcal J$) called pieces such that
 $X=\cup_{i\in\mathcal J}S_i$, and that $S_i\cap\bar{S_j}\neq\emptyset\Leftrightarrow S_i\subset\bar{S_j}\Leftrightarrow i\leq j$.
  We call the space $X$ a $\mathcal J$- {\bf decomposed space}.

A stratified space is defined recursively as follows.
 A  decomposed space $X$ is called a {\bf stratified space} if the pieces of $X$, called strata, satisfy the following condition:
Given a point $x$ in a piece $S$ (connected),  there exists an open neighborhood $\tilde{U}$ of $x$ in $X$, an open ball $B$ around $x$ in
  $S$, a compact stratified space $L$, called the {\bf link of x}, and a homeomorphism
   $$\varphi: B\times \overset{\circ}{C}L\rightarrow \tilde{U}$$
  that preserves the decompositions. Here, $\overset{\circ}{C}L$ is the space obtained by collapsing the boundary $L\times {0}$ of the
  half-open cylinder $L\times [0, \infty)$ to a point.
 We also call the  link of $x$ the {\bf link of S}.

    For a smooth manifold $M$, if $N$ is a closed submanifold of $M$, by the tubular neighborhood theorem, a neighborhood of $N$ in $M$
is diffeomorphic to a neighborhood of the normal bundle $E$ of $N$ in $M$. If the fiber sphere of the sphere bundle $S(E)$ of $N$ is
connected and simply connected, i.e., if the codimension of $N$ is bigger than or equal to $3$, then $\pi_1(M)\cong\pi_1(M-N)$.
For a stratified space $X$, the {\em link} of a stratum $S$  plays the role of the fiber sphere of the sphere bundle of a submanifold in
a smooth manifold. Using the Van-Kampen theorem, we can prove the following lemma. May see Lemma 2.3 in \cite{L2} for detail.
\begin{lemma}\label{remove:str}
Let $X$ be a stratified space. Let $S$ be a connected closed stratum of $X$ whose link is connected and simply connected.
Then $\pi_1(X)\cong\pi_1\left(X- S\right)$.
\end{lemma}



\section{Proof of Proposition~\ref{M=M/G}}

  \begin{lemma}\label{bredon}
    (Corollary 6.3 in \cite{Br}) If $X$ is an arcwise connected $G$-space, $G$ compact Lie,
    and if there is an orbit which is connected, then
    the fundamental group of $X$ maps onto that of $X/G$.
    \end{lemma}


 Let $M$ be a connected smooth $G$-manifold, where $G$ is a connected compact
 Lie group. In the Introduction, we defined the $(H)$-orbit type $M_{(H)}$ for a
 stabilizer group $H$ and we defined the principal orbit type. The fact is that
 all other orbit types are in the closure of
 the principal orbit type. More generally, if $M_{(H)}\subset \bar M_{(H')}$, then $(H')\subset (H)$.
The quotient $M/G$ is a stratified space whose strata are the
$M_{(H)}/G$'s. The manifold $M_{(H)}/G$ is called the
 {\bf $(H)$-stratum} of $M/G$.






\begin{lemma}\label{homotopic}
 Let $G$ act on a connected smooth manifold $M$, where $G$ is a connected compact Lie group.
Let $M_{(H)}$ be a closed connected orbit type which has codimension
at least $2$ in $M$. Then we have a surjection
$$\pi_1\left(M- M_{(H)}\right)\twoheadrightarrow\pi_1\left(M\right).$$
\end{lemma}
\begin{proof}
Since the submanifold $M_{(H)}$ has codimension at least $2$ in $M$,
its sphere bundle $S\left(M_{(H)}\right)$ (in particular the fiber)
is connected, by the homotopy exact sequence for the fibration
$S\left(M_{(H)}\right)\to M_{(H)}$, we have a surjection
$$\pi_1\left(S\left(M_{(H)}\right)\right)\twoheadrightarrow \pi_1\left(M_{(H)}\right).$$
Take the open cover $\{M- M_{(H)}, N\left(M_{(H)}\right) \}$ of $M$,
where  $N\left(M_{(H)}\right)$ is a small open neighborhood of
$M_{(H)}$. By the Van-Kampen theorem, we have a surjection
$$\pi_1\left(M- M_{(H)}\right)\twoheadrightarrow \pi_1\left(M\right).$$
\end{proof}

\begin{remark}
Let $G$ act on a connected smooth manifold $M$, where $G$ is a
connected compact Lie group. Assume that {\em each} nonprincipal
orbit type has codimension at least $2$ in $M$. Using
Lemma~\ref{homotopic} inductively, in a suitable order, we can
continue to throw away certain orbit types to get a surjection. For
example, if $\mathcal M_{(H)}$ is the submanifold which consists of
those orbit types whose closures contain $M_{(H)}$, where $M_{(H)}$
is a closed connected orbit type, then
$$\pi_1\left(\mathcal M_{(H)}\right)\twoheadrightarrow\pi_1\left(M\right);$$
and, if $M_P$ is the principal orbit type, then we have a surjection
$$\pi_1\left(M_P\right)\twoheadrightarrow \pi_1\left(M\right).$$
\end{remark}


\begin{proof} [Proof of Proposition~\ref{M=M/G}]
Since $G$ is connected, by  Lemma~\ref{bredon}, we have a surjection
$$\pi_1(M)\twoheadrightarrow\pi_1\left(M/G\right).$$ 

Let $G\cdot x$ be the simply connected orbit. Let $q\colon M\to M/G$
be the quotient map and let $\bar x=q(x)$.
 Let $\alpha$ be a loop in $M/G$ based at $\bar x$ such that $[\alpha]=1\in\pi_1\left(M/G; \bar x\right)$.
 Let $\Tilde \alpha$  be a loop in $M$ based at $x$ which is any lift of $\alpha$.
 We need to show that  $[\Tilde\alpha]=1\in\pi_1\left(M;  x\right)$.


 First note that in each orbit type, the orbits are diffeomorphic to
 each other, so we can use this diffeomorphism to deform paths or loops in
 the same orbit type. Secondly, for a nonprincipal orbit type, since
 its codimension in $M$ is at least $2$,
  by Lemma~\ref{homotopic}, we can push paths or loops out of
 this nonprincipal orbit type to its sphere bundle. For the loop
 $\Tilde\alpha$, we need to deform it through a family of base point
 preserving loops to the constant loop $x$. Since
$[\alpha]=1\in \pi_1\left(M/G; \bar x\right)$, we have a continuous
family of paths from each point of $\alpha$ to $\bar x$. In $M$, we
follow this continuous family of paths between the orbits which
intersect $\Tilde\alpha$ and the orbit $G\cdot x$, and we use the
above two types of deformation to deform $\Tilde\alpha$ to a loop
$\Tilde\alpha'$ which lies in the orbit $G\cdot x$. Since
$\pi_1\left(G\cdot x\right)=1$, $[\Tilde\alpha]=[\Tilde\alpha']=1\in
\pi_1\left(M; x\right)$.
\end{proof}



\begin{remark}
Using Proposition~\ref{M=M/G} on a subspace $\mathcal M_{(H)}$ as
follows, we have
\begin{lemma}
 Let $G$, a connected compact Lie group, act on a connected  smooth manifold $M$.
 Let $M_{(H)}$ be a connected orbit type whose orbits are simply connected. Let $\mathcal M_{(H)}$
be the submanifold which consists of those orbit types whose
closures contain $M_{(H)}$. Assume that each nonprincipal orbit type
in $\mathcal M_{(H)}$ has codimension at least $2$ in $\mathcal
M_{(H)}$.  Then
$$\pi_1\left(\mathcal M_{(H)}\right)\cong\pi_1\left(\mathcal M_{(H)}/G \right).$$
\end{lemma}
\end{remark}



\section{Proof  of Theorem~\ref{thm2}}

\begin{lemma}\label{codim}
Let $M$ be a connected Hamiltonian $G$-manifold, where $G$ is a connected compact Lie group.
Then,  each nonprincipal orbit type
 has codimension at least $2$.
\end{lemma}
\begin{proof}
First assume that $G=T$ is abelian. By the local normal form
theorem, a neighborhood of an orbit with isotropy group $H$ is
isomorphic to $A=T\times_H (\mathfrak h^{\circ}\times V)=T\times_H
(\mathfrak h^{\circ}\times W\times V^H)$, where $V^H$ is the
subspace of $V$ fixed by $H$ and $W$ is its complementary subspace
in $V$. Note that $A_{H}=T\times_H (\mathfrak h^{\circ}\times
V^H)$. So if $H$ is not the principal stabilizer, then $W$ is a nontrivial
complex vector space on which $H$ acts nontrivially; so $M_{H}$ has codimension
at least $2$.

   Now assume that $G$ is nonabelian. Fix an isotropy type $(H)$. We may assume that
 the image of the orbit type $M_{(H)}$ intersects a face $\tau$ of $\mathfrak t^*_+$ (then $H\subset G_{\tau}$).
By equivariance of the moment map, we only need to consider this
   orbit type in the cross section $R$ on which  $G_{\tau}$ acts.  By the local normal form
   theorem, a neighborhood of an  isotropic orbit in $R$ with stabilizer $(H)$ is isomorphic to
    $A=G_{\tau}\times_H (\mathfrak h^{\circ}\times V)=G_{\tau}\times_H
(\mathfrak h^{\circ}\times W\times V^H)$, where $\mathfrak h^{\circ}$ is the annihilator of $\mathfrak h=$Lie$(H)$ in
$\mathfrak g^*_{\tau}$. Then $A_{(H)}=G_{\tau}\times_H
(\R^m\times V^H)$, where $\R^m$ is a subspace of $\mathfrak
h^{\circ}$ fixed by $H$. Let $\R^l$ be the complementary subspace of
$\R^m$ in $\mathfrak h^{\circ}$.  If $(H)$ is not the principal stabilizer type, then $\R^l\oplus
W\neq\emptyset$.  Based on Proposition 6.3 in \cite{L2} (one may refer to the proof of Lemma 6.19 in \cite{L2} for
all possible $H$), $\dim(\R^l\oplus W)\geq 2$.
\end{proof}

\begin{lemma}\label{G/T}
Let $G$ be a connected compact Lie group, and let $T$ be a maximal
torus of $G$. Let $H$ be a subgroup of $G$ such that $T\subseteq H$.
Then $\pi_1\left(G/H\right)=1$.
\end{lemma}

\begin{proof}
Since we have a surjection
$$\pi_1(T)\twoheadrightarrow\pi_1(G)\quad\mbox{and the inclusions}$$
$$T\overset{\iota}\hookrightarrow H\overset{\iota}\hookrightarrow G,$$
we have a surjection
$$\pi_1(H)\twoheadrightarrow\pi_1(G).$$
The claim follows from the homotopy exact sequence for the fibration
$H\subset G\to G/H$:
$$\cdots \to\pi_1(H)\to \pi_1(G)\to\pi_1\left(G/H\right)\to\cdots .$$
\end{proof}




\begin{proposition}\label{levelset}
Let $M$ be a connected Hamiltonian $G$-manifold with proper
equivariant moment map $\phi$, where $G$ is a connected compact Lie
group.
 Let $T$ be a maximal torus of $G$.
Assume that there is a subgroup $H$ of $G$ such that $H\supseteq T$
and that $H$ has a fixed point $x$, and $\phi(x)=a$. Then
$$\pi_1\left(\phi^{-1}(G\cdot a)\right)\cong\pi_1\left(\phi^{-1}(a)\right)\cong\pi_1\left(M_a \right).$$
\end{proposition}
\begin{proof}
Since $\phi$ is proper, $\phi^{-1}(G\cdot a)$ and $\phi^{-1}(a)$ are
connected by Theorem~\ref{convexity}; and, by definition of
properness, $\phi^{-1}(G\cdot a)$ and $\phi^{-1}(a)$  are also
compact. Hence,  $M_a$ is connected and compact.



Consider the map
\begin{equation}\label{level}
\pi_1\left(\phi^{-1}(G\cdot a)\right)\to\pi_1\left(M_a \right).
\end{equation}



 Since $G$ is connected and it acts on $\phi^{-1}(G\cdot a)$, by
Lemma~\ref{bredon},  (\ref{level}) is onto.


 Now we show that (\ref{level}) is injective.  Let $\mathcal N$ be an open invariant neighborhood
 of $\phi^{-1}(G\cdot a)$ in $M$ which equivariantly deformation retracts to $\phi^{-1}(G\cdot a)$.
Since the orbit $G\cdot x\approx G/H$, by Lemma~\ref{G/T},
$\pi_1\left(G\cdot x\right)=1$. By Lemma~\ref{codim}, each
nonprincipal orbit type in $\mathcal N$ has codimension at least
$2$. Take $[\alpha]=1\in\pi_1\left(M_a\right)$. Then
$[\alpha]=1\in\pi_1\left(\mathcal N/G\right)$. Let $\Tilde
\alpha\subset\phi^{-1}(G\cdot a)$  be a lifted loop of $\alpha$. By
Proposition~\ref{M=M/G}, $[\Tilde \alpha]=1\in\pi_1\left(\mathcal
N\right)$.  Since $\pi_1\left(\mathcal
N\right)=\pi_1\left(\phi^{-1}(G\cdot a)\right)$, $[\Tilde
\alpha]=1\in\pi_1\left(\phi^{-1}(G\cdot a)\right)$.


 Now, we prove that $\pi_1\left(\phi^{-1}(a)\right)\cong\pi_1\left(M_a \right)$. Assume $a\in\tau\subset\mathfrak t^*_+$.
Since $H$ fixes $x$, and since
$\phi(g\cdot x)=Ad^*(g)\cdot\phi(x)$, $H\subset G_a$. Consider the $G_a$ action on the cross section $R^{\tau}$ and on
$\phi^{-1}(a)$. Take a $G_a$-invariant open neighborhood of $\phi^{-1}(a)$ in $R^{\tau}$ which $G_a$- equivariantly deformation
retracts to $\phi^{-1}(a)$ and use the
same argument as above.
\end{proof}





\begin{proof} [Proof  of Theorem~\ref{thm2}]
(A). Let $K$ be $G$, $T$ or $H$. Then $K$ is connected and it acts
on $M$. Since $M$ is a Hamiltonian $G$-manifold, it is a Hamiltonian
$K$-manifold for any $K$. Moreover, the $K$ action has an orbit
which is simply connected by Lemma~\ref{G/T}. Then
$\pi_1\left(M\right)\cong\pi_1\left(M/K\right)$ follows from
Lemmas~\ref{bredon}, \ref{codim} and Proposition~\ref{M=M/G}.

(B) is Proposition~\ref{levelset}.
\end{proof}



 \section{Proof of (A) of Theorem~\ref{thm3}}\label{sectionM/G}

 The method of removing and deforming used in \cite{L2} to prove the same fact for compact manifold
  $M$ applies to this case. Hamiltonian action plays crucial roles
 in the removing process --- we have the local normal form theorem and the cross section theorem. The {\em properness of $\phi$}
 or the {\em compactness of $M$}  is used to control
 the gradient flow of suitable components of $\phi$ or
  of $\|\phi\|^2$  --- we use these flows to do {\em local} deformation retractions (Theorem~\ref{retract}).
  We will state the important lemmas which lead
 to the results and we will emphasize the key points which make these lemmas hold. The corresponding proofs in
 \cite{L2} apply or apply with slight modifications.

   
    \begin{lemma}\label{a's} (see Lemma 3.6 in \cite{L2})
   Let $(M, \omega)$ be a connected  Hamiltonian $T$- manifold
      with proper moment map $\phi$, where $T$ is a connected compact torus.
Then, for all value $a$'s  in one connected  chamber of
im$(\phi)$, $\pi_1(M_a)$'s are isomorphic.
    \end{lemma}

For the method of removing, the results in Lemmas~\ref{Tlink0} and \ref{Glink0} below are the key reasons for Lemmas 7.20, 3.8 and
6.19 in \cite{L2} to hold. Lemmas 7.20, 3.8 and 6.19 in \cite{L2} were stated in emphasizing the consequences of these results.
Here and in Lemmas~\ref{Tlinksame} and \ref{Glinksame}, we are particularly emphasizing these points themselves.

\begin{lemma}\label{Tlink0} (see Lemmas 7.20 and 3.8 in \cite{L2})
 Let $M$ be a connected Hamiltonian $T$-manifold with proper moment map $\phi$, where $T$ is a connected compact torus.
 Let $\mathcal F$ be a singular face of im$(\phi)$.  Let $\bar U$ be the closure of
one open connected chamber such that $\mathcal F \subset\bar U$.
\begin{itemize}
\item [(1)] For each {\em nonprincipal isotropy type}
 $M_{(H)}\cap\phi^{-1}(\mathcal F)$,
the link of  $\left(M_{(H)}\cap \phi^{-1}(\mathcal F)\right)/T$  in
$M/T$ is connected and simply connected.
\item [(2)]
For each isotropy type  $M_{(H)}\cap \,\phi^{-1}(\mathcal F)$, the
link of  $\left(M_{(H)}\cap\phi^{-1}(\mathcal F)\right)/T$  in
$\phi^{-1}(\bar U)/T$ is connected and simply connected.
\end{itemize}
\end{lemma}


We can use (2) of Lemma~\ref{Tlink0} to prove Lemma~\ref{a=c}. The idea is to use Lemma~\ref{remove:str} and Theorem~\ref{retract}.
We refer to the proof of Lemma 3.8 in \cite{L2}.

    \begin{lemma}\label{a=c}
        Let $(M, \omega)$ be a connected  Hamiltonian $T$-manifold
      with proper moment map $\phi$, where $T$ is a connected compact torus.
      Let $c$ be a value on a singular face of im$(\phi)$, and let
      $a$ be a regular value very near $c$.
      Then
      $\pi_1(M_c)\cong\pi_1(M_a)$.
     \end{lemma}

    \begin{lemma}\label{G-a's}(see Lemma 6.15 in \cite{L2})
         Let $(M, \omega)$ be a connected Hamiltonian  $G$-manifold
  with proper moment map $\phi$, where $G$
   is a connected compact non-abelian Lie group. Let  $\tau^P\subset\mathfrak t^*_+$
   be the principal face. Then, for all value $a$'s on
    $\tau^P$, $\pi_1(M_a)$'s are isomorphic.
  \end{lemma}

\begin{lemma}\label{Glink0} (see Lemma 6.19 in \cite{L2})
 Let $M$ be a connected Hamiltonian $G$-manifold with proper moment map $\phi$,
 where $G$ is a connected compact non-abelian Lie group.
Let $C$ be the central face of $\mathfrak t^*_+$, and assume that
$C\cap\mbox{im}(\phi)\neq \emptyset$ and that $C$ is not the only
face of $\mathfrak t^*_+$ which contains $\mbox{im}(\phi)$.
 Then, for each isotropy type  $M_{(H)}\cap\phi^{-1}(C)$,
the link of  $\left(M_{(H)}\cap\phi^{-1}(C)\right)/G$  in  $M/G$
is connected and simply connected.
\end{lemma}

 We can use Lemma~\ref{Glink0} to prove Lemma~\ref{G-a=c} below. See
 the proof of Lemma 6.16 in \cite{L2}.
   \begin{lemma}\label{G-a=c}
         Let $(M, \omega)$ be a connected Hamiltonian  $G$-manifold
  with proper moment map $\phi$, where $G$
   is a connected compact non-abelian Lie group.
    Let $a$ be a generic value on the  principal face
    $\tau^P$ of $\mathfrak t^*_+$, and let $c$ be a value which is not on
   $\tau^P$. Then
 $\pi_1(M_c)\cong\pi_1(M_a)$.
   \end{lemma}


 Lemmas~\ref{a's},   \ref{a=c}, \ref{G-a's} and  \ref{G-a=c}   imply Proposition~\ref{M_a}.
 \begin{proposition}\label{M_a}
      Let $(M, \omega)$ be a connected Hamiltonian $G$-manifold
      with proper moment map $\phi$, where $G$
   is a connected compact Lie group.
   Then,  $\pi_1(M_a)$'s are isomorphic for
   all $a\in \mbox{im}(\phi)$.
  \end{proposition}





  Similar to the proof of Theorem 1.6 in \cite{L2}, we can use Lemmas~\ref{Tlink0} and \ref{Glink0}  to prove
 Proposition~\ref{M/G} which is (A) of Theorem~\ref{thm3}.
 \begin{proposition}\label{M/G}
      Let $(M, \omega)$ be a connected Hamiltonian $G$-manifold
      with proper  moment map $\phi$, where $G$
   is a connected compact Lie group.
   Then,  $\pi_1(M/G)\cong\pi_1(M_a)$  for
   all $a\in \mbox{im}(\phi)$.
  \end{proposition}



 \section{Proof of (B) of Theorem~\ref{thm3} for $G=T$}


\begin{lemma}\label{Tisom}
 Let $(M, \omega)$ be a connected Hamiltonian $T$-manifold
with proper moment map $\phi$, where $T$ is a connected compact torus.
Let $T'\subset T$ be a connected subgroup.
Then, for all values $a$'s  in one connected  chamber of
im$(\phi)$, $\pi_1\left(\phi^{-1}(a)/T'\right)$'s are isomorphic.
\end{lemma}
\begin{proof}
This is because  for all values $a$'s  in one connected  chamber of
im$(\phi)$, $\phi^{-1}(a)$'s are $T$-equivariantly diffeomorphic.
\end{proof}

\begin{lemma}\label{Tlinksame}
 Assume the assumptions of Theorem~\ref{thm3} hold, where $G=T$ is a connected compact torus.
Let $\Th\subset T$  be a connected subgroup such that up to a finite subgroup,
every isotropy group is a subgroup of $\Th$.
Let $\mathcal F$ be a singular face on im$(\phi)$.
Let $\bar U$ be the closure of one open connected chamber $U$ such
that $\mathcal F \subset\bar U$. Let $N$ be $M$ or be $\phi^{-1}(\bar U)$.
For each isotropy type $M_{H}\cap\phi^{-1}(\mathcal F)$,
 let $L_H$ be the link of
$\left(M_{H}\cap\phi^{-1}(\mathcal F)\right)/T$  in $N/T$, and let $L'_H$ be the link of
$\left(M_{H}\cap\phi^{-1}(\mathcal F)\right)/\Th$  in $N/\Th$.
 \begin{itemize}
 \item  If $H \subseteq \Th$, then $L_H=L'_H$.
 \item  If $H\subseteq  \Th$ except for a finite subgroup
 $\Gamma$ of $H$, then $L_H=L'_H/\Gamma$, and, $L_H$ and $L'_H$ are homeomorphic.
\end{itemize}
\end{lemma}
\begin{proof}
Let us prove the claims for $N=M$.
Let $p\in M$ be a point with stabilizer $H$ such that $\phi(T\cdot
p)\in\mathcal F$. By the local normal form theorem (Theorem~\ref{form}), a neighborhood
of the (isotropic) orbit $T\cdot p$ in $M$ is isomorphic to
$A=T\times_H(\mathfrak h^{\circ}\times V)=T\times_H(\mathfrak
h^{\circ}\times W\times V^H)$, where $\mathfrak h=$Lie$(H)$,
$\mathfrak h^{\circ}$ is its annihilator in $\mathfrak t^*$, $V^H$
is the subspace of $V$ fixed by $H$ and $W$ is such that $V=W\oplus
V^H$. The moment map $\phi$ on $A$ is $\phi ([t, a, w,
v])=\phi(p)+ a + \psi(w)$, where $\psi$ is the  moment map of
the $H$ action on $W$. The isotropy type
$A_H\cap\phi^{-1}(\mathcal F)=T\times_H(\R^m\times 0\times V^H)$,
where $\R^m\subset\mathfrak h^{\circ}$ is a subspace which is mapped to
$\mathcal F$. Let $\R^l$ be the subspace such that $\mathfrak
h^{\circ}=\R^l\oplus\R^m$ (note that $H$ acts on $\mathfrak h^{\circ}$ by the trivial coadjoint action).

We see that $\left(A_H\cap\phi^{-1}(\mathcal F)\right)/T=\R^m\times 0\times V^H$, and
$A/T=\mathfrak h^{\circ}\times V^H\times W/H$. So $L_H=S(\R^l\times
W)/H$. Here, $S(\R^l\times W)$ denotes a sphere of the vector space $\R^l\times W$.

If $H\subseteq \Th$,  then
$\left(A_H\cap\phi^{-1}(\mathcal F)\right)/\Th=T/\Th\times\R^m\times
0\times V^H$, and $A/\Th=T/\Th\times\mathfrak
h^{\circ}\times V^H\times W/H$. So $L'_H=S(\R^l\times W)/H$.

If $H\subseteq \Th$ except for a finite subgroup
 $\Gamma$, then
$\left(A_H\cap\phi^{-1}(\mathcal F)\right)/\Th
=T/\Th\times_H\left(\R^m\times 0\times
V^H\right)=T/\Th\times_{\Gamma}\left(\R^m\times V^H\right)$,
and $A/\Th = T/\Th\times_H \left(\mathfrak h^{\circ}\times
V^H\times W\right)=(T/\Th\times\R^m\times
V^H)\times_{\Gamma}\left((\R^l\times W)/(H\cap \Th)\right)$. So
$L'_H=S(\R^l\times W)/(H\cap \Th)$. Therefore,
$L_H=L'_H/\Gamma=S(\R^l\times W)/(H\cap \Th\times\Gamma)$. Since
$\Gamma$ acts on $\R^l$ trivially and it acts on $W$ as a subgroup
of $T=(S^1)^m$ (for some $m$), we easily see that $L_H$ and $L'_H$
are homeomorphic.

We can similarly prove the claims for $\bar U$. We refer to the proof of Lemma 3.8 in \cite{L2}
for the computation of $L_H$. Then one can compute $L'_H$ as above.
\end{proof}



\begin{lemma}\label{Tlink}
 Assume the assumptions of Theorem~\ref{thm3} hold,  where $G=T$ is a connected compact torus.
Let $\Th\subset T$  be a connected subgroup such that up to a finite subgroup,
every isotropy group is a subgroup of $\Th$.
 Let $\mathcal F$ be a singular face on im$(\phi)$.  Let $\bar U$ be the
closure of one open connected chamber $U$ such that $\mathcal F
\subset\bar U$.
\begin{itemize}
\item [(1)] For each {\em nonprincipal  isotropy type}
$M_{(H)}\cap\phi^{-1}(\mathcal F)$,
the link of $\left(M_{(H)}\cap\phi^{-1}(\mathcal F)\right)/\Th$ in
$M/\Th$ is connected and simply connected.
\item [(2)]  For each isotropy type $M_{(H)}\cap\,\phi^{-1}(\mathcal F)$, the link of  $\left(M_{(H)}\cap\phi^{-1}(\mathcal F)\right)/\Th$
in $\phi^{-1}(\bar U)/\Th$ is connected and simply connected.
\end{itemize}
\end{lemma}
\begin{proof}
This follows from Lemmas~\ref{Tlink0} and \ref{Tlinksame}.
\end{proof}




     \begin{lemma}\label{T-remove}
  Assume the assumptions of Theorem~\ref{thm3} hold, where $G=T$ is a connected compact torus.
Let $\Th\subset T$  be a connected subgroup such that up to a finite subgroup,
every isotropy group is a subgroup of $\Th$.
 Let $c$ be a singular value, and let $a$ be a regular value very near $c$. Let $O$ be a small open neighborhood of $c$ containing $a$. Let $O'$ be the intersection of $O$ with the connected open chamber containing $a$, and let
$\bar{O'}$ be its closure in $O$. Let $B$ be the set of values in $\bar{O'}$ but not in $O'$.
 Then
$$\pi_1\left(\phi^{-1}(\bar{O'})/\Th\right)\cong\pi_1\left(\phi^{-1}(\bar{O'})/\Th-\phi^{-1}(B)/\Th\right).$$
   \end{lemma}
 \begin{proof}
 We inductively use (2) of Lemma~\ref{Tlink} to remove $\phi^{-1}(B)/\Th$ from $\phi^{-1}(\bar{O'})/\Th$.
 One may refer to the proof of Lemma 3.8 in \cite{L2}.
  \end{proof}


\begin{lemma}\label{Tisom'}
     Assume the assumptions of Theorem~\ref{thm3} hold, where $G=T$ is a connected compact torus.
      Let $c$ be a singular value, and let
      $a$ be a regular value very near $c$.
      Then
      $\pi_1\left(\phi^{-1}(c)/\Th\right)\cong\pi_1\left(\phi^{-1}(a)/\Th\right)$.
     \end{lemma}
\begin{proof}
 We take the $O$ in Lemma~\ref{T-remove} small enough so that $\phi^{-1}(O)$ and $\phi^{-1}(\bar{O'})$
 equivariantly
 deformation retracts to $\phi^{-1}(c)$ (Theorem~\ref{retract}). So
 $$\pi_1\left(\phi^{-1}(\bar{O'})/\Th\right)\cong\pi_1\left(\phi^{-1}(c)/\Th\right).$$
Since
$$\pi_1\left(\phi^{-1}(\bar{O'})/\Th-\phi^{-1}(B)/\Th\right)\cong\pi_1\left(\phi^{-1}(a)/\Th\right),$$
the claim follows from Lemma~\ref{T-remove}.
\end{proof}



\begin{lemma}\label{TT_m}
 Assume the assumptions of Theorem~\ref{thm3} hold, where $G=T$ is a connected compact torus.
Let $\Th\subset T$  be a connected subgroup such that up to a finite subgroup,
every isotropy group is a subgroup of $\Th$.
Let $\mathcal{F}$ be a singular face on im$(\phi)$.
Let $O$ be a small open neighborhood of $\mathcal{F}$ on the moment map image ($O$ does not intersect the faces which are in the closure of $\mathcal{F}$). Let $S$ be the set of nonprincipal orbits  in $\phi^{-1}(\mathcal{F})$.
Then $\pi_1\left(\phi^{-1}(O)/\Th\right)\cong\pi_1\left(\phi^{-1}(O)/\Th-S/\Th\right)$.
\end{lemma}
\begin{proof}
The argument is similar to the proof of Lemma 7.20 in \cite{L2}, but using (1) of Lemma~\ref{Tlink}.
\end{proof}

\begin{proof}[Proof  of (B) of Theorem~\ref{thm3} for $G=T$.]
Using Lemmas~\ref{Tisom}, \ref{Tisom'} and \ref{TT_m}, follow the argument used in the proof of Theorem 1.6 in \cite{L2}
for the case $G=T$.
\end{proof}



 \section{Proof of (B) of Theorem~\ref{thm3} for nonabelian $G$}

\begin{lemma}\label{Glinksame}
Assume the assumptions of Theorem~\ref{thm3} hold, where $G$ is nonabelian.
Let $C$ be the central face of $\mathfrak t^*_+$, and
assume that $C\cap\mbox{im}(\phi)\neq \emptyset$ and that $C$ is not
the only face of $\mathfrak t^*_+$ which contains $\mbox{im}(\phi)$.
 For each isotropy type $M_{(H)}\cap\phi^{-1}(C)$,
 let $L_H$ be the link of
$\left(M_{(H)}\cap\phi^{-1}(C)\right)/G$  in $M/G$, and let $L'_H$ be the link of  $\left(M_{(H)}\cap\phi^{-1}(C)\right)/\Gh$
in $M/\Gh$.
\begin{itemize}
\item  If $H\subseteq \Gh$, then $L_H=L'_H$, they are connected and $\pi_1(L_H)=\pi_1(L'_H)=1$.
\item  If $H\subseteq \Gh$ except for finitely many elements, then
 $L_H$ and $L'_H$ are connected and $\pi_1(L_H)=\pi_1(L'_H)=1$.
 \end{itemize}
\end{lemma}
 \begin{proof}
By the local normal form theorem (Theorem~\ref{form}), a neighborhood in $M$ of an orbit
$\mathrm O$ in $\phi^{-1}(C)$ with isotropy group
  $(H)$ is isomorphic to
$A=G\times_H (\mathfrak{h}^{\circ}\times V)=G\times_H
(\mathfrak{h}^{\circ}\times W\times V^H)$, where
$\mathfrak{h}^{\circ}$ is the annihilator of
 $\mathfrak{h}=$Lie$(H)$ in $\mathfrak{g}^*=$Lie$^*(G)$, and,
$V^H$ is the subspace of $V$ fixed by $H$ and $W$ is such that
$V=W\oplus V^H$. The moment map $\phi$ on $A$ is $\phi ([g, a, w,
v])=Ad^*(g)\left(\phi(\mathrm O)+ a + \psi(w)\right)$, where $\psi$
is the moment map of the $H$ action on $W$. So
$A_{(H)}\cap\phi^{-1}(C) = G\times_H(\R^m\times 0\times V^H)$, where
$\R^m\subset\mathfrak h^{\circ}$ is the subspace which is mapped to $C$
and therefore on which $H$ acts trivially.  Let $\R^l\subset\mathfrak h^{\circ}$ be the subspace such that
$\mathfrak h^{\circ}=\R^l\oplus\R^m$.


Then $\left(A_{(H)}\cap\phi^{-1}(C)\right)/G=\R^m\times 0\times V^H$, and
$A/G=\R^m\times 0\times V^H\times (\R^l\times W)/H$. So
$L_H=S(\R^l\times W)/H$.

If $H\subseteq \Gh$, then
$\left(A_{(H)}\cap\phi^{-1}(C)\right)/\Gh=G/\Gh\times\R^m\times V^H$, and
$A/\Gh = G/\Gh\times\R^m\times V^H \times (\R^l\times W)/H$.
So $L'_H=S(\R^l\times W)/H=L_H$.

If $H\subseteq \Gh$ except for finitely many elements of $H$, then
$\left(A_{(H)}\cap\phi^{-1}(C)\right)/\Gh$  $=G/\Gh\times_H (\R^m\times V^H)=(\R^m\times V^H)\times G/\Gh\times_H 0$, and
$A/\Gh=G/\Gh\times_H (\R^m\times V^H\times \R^l\times
W)=(\R^m\times V^H)\times G/\Gh\times_H (\R^l\times W).$ So
$L'_H=S(\R^l\times W)/(H\cap \Gh)$.

   By Lemma~\ref{Glink0},
$L_H$ is connected and $\pi_1(L_H)=1$. The claim for the first case when $L_H=L'_H$ follows immediately. For the second case,
we think about it as follows. If we take the new subgroup $H'=H\cap \Gh$ of $G$ which differs from $H$ by finitely many
elements, the $S(\R^l\times W)$ in the above analysis does not change; by the same arguments (or criteria) of the connectedness and simply connectedness of $L_H$, $L'_H$ is connected and simply connected.
For details of the arguments, we refer to the proof of Lemma 6.19 in \cite{L2}.
\end{proof}


\begin{lemma}\label{Glink}
  Assume the assumptions of Theorem~\ref{thm3} hold, where $G$ is nonabelian.
 Let $C$ be the central face of $\mathfrak t^*_+$, and
assume that $C\cap\mbox{im}(\phi)\neq \emptyset$ and that $C$ is not
the only face of $\mathfrak t^*_+$ which contains $\mbox{im}(\phi)$.
Then
$$\pi_1\left (M/\Gh\right )\cong \pi_1\left (M/\Gh-\phi^{-1}(C)/\Gh\right ).$$
Similarly, if $O$ is a small open invariant  neighborhood of $C$ in
$\mathfrak g^*$, then
$$\pi_1\left (\phi^{-1}(O)/\Gh\right )\cong \pi_1\left (\phi^{-1}(O)/\Gh-\phi^{-1}(C)/\Gh\right ).$$
\end{lemma}
\begin{proof}
By Lemma~\ref{Glinksame}, for each possible $H$, the link of $\left(M_{(H)}\cap\phi^{-1}(C)\right)/\Gh$ in $M/\Gh$
is connected and simply connected.
Therefore we can inductively remove the strata of
$\phi^{-1}(C)/\Gh$ from $M/\Gh$ or from
$\phi^{-1}(O)/\Gh$ using Lemma~\ref{remove:str}.
\end{proof}


\begin{lemma}\label{link}
  Assume the assumptions of Theorem~\ref{thm3} hold, where $G$ is nonabelian.
\begin{enumerate}
\item  Let $\tau$ be a non-principal
  face of  $\mathfrak t^*_+$  such that
  $\tau\cap\mbox{im}(\phi)\neq\emptyset$ and let $R^{\tau}$ be the corresponding cross section. Then, for each
 $M_{(H)}\cap\phi^{-1}(\tau)$, the link of
  $\left(M_{(H)}\cap\phi^{-1}(\tau)\right)/(G_{\tau}\cap \Gh)$ in $R^{\tau}/(G_{\tau}\cap \Gh)$ is the same as the link of  $\left(M_{(H)}\cap\phi^{-1}(G\cdot\tau)\right)/\Gh$ in $G\cdot R^{\tau}/\Gh$.
\item  Let $\tau^P$ be the principal face and let $R^{\tau^P}$
be the principal cross section. Let
 $c\in\tau^P$ be a singular value. Then, for each  $M_{(H)}\cap\phi^{-1}(c)$, the link of
$\left(M_{(H)}\cap\phi^{-1}(c)\right)/(T\cap \Gh)$ in
 $R^{\tau^P}/(T\cap \Gh)$ is the same
 as the link of  $\left(M_{(H)}\cap\phi^{-1}(G\cdot c)\right)/\Gh$ in $G\cdot R^{\tau^P}/\Gh$.
\end{enumerate}
\end{lemma}
\begin{proof}
 First note that, if a point with isotropy group $H$ is mapped to a face $\tau$, then since $\phi (g\cdot m)=Ad^*(g)\cdot\phi(m)$,
$H\subset G_{\tau}$.

(1) The manifold $G\cdot R^{\tau}$ is a fibration over the coadjoint orbit $G/G_{\tau}$ with fiber $R^{\tau}$; and
correspondingly, each  $M_{(H)}\cap (G\cdot R^{\tau})$ is a fibration over $G/G_{\tau}$ with fiber
$M_{(H)}\cap R^{\tau}$, moreover,  each $M_{(H)}\cap\phi^{-1}(G\cdot \tau)$ is a fibration over $G/G_{\tau}$ with fiber
$M_{(H)}\cap\phi^{-1}(\tau)$.  The claim follows from the equivariance of $\phi$.

The proof of (2) is similar.
\end{proof}

\begin{remark}
Note that for any face $\tau\subset\mathfrak t^*_+$ and the stabilizer group $G_{\tau}$ of $\tau$,
the subgroup $G_{\tau}\cap \Gh$ of $G_{\tau}$  is connected  since $G_{\tau}$ and $\Gh$ are both connected.
\end{remark}


\begin{lemma}\label{Gisom}
   Assume the assumptions of Theorem~\ref{thm3} hold, where $G$ is nonabelian.   Let  $\tau^P\subset\mathfrak t^*_+$
   be the principal face. Then,
 $\pi_1\left(\phi^{-1}(G\cdot\tau^P)/\Gh\right)\cong\pi_1\left(\phi^{-1}(G\cdot a)/\Gh\right)$ for
  all $a\in\tau^P$.
  \end{lemma}
\begin{proof}
First, consider the space $\phi^{-1}(\tau^P)$ with the $T$ action,
where $T$ is a maximal torus of $G$. Similar to the proof of the
theorem for abelian group actions, by using removing and deforming
in the space $\phi^{-1}(\tau^P)/(\Gh\cap T)$, we get
$$\pi_1\left(\phi^{-1}(\tau^P)/(\Gh\cap T)\right)\cong\pi_1\left(\phi^{-1}(a)/(\Gh\cap
T)\right)$$
for some {\em particular} $a\in \tau^P$.
Then by Lemma~\ref{Tlink} and (2) of  Lemma~\ref{link},  we can  do corresponding removing and deforming in the space
$\phi^{-1}(G\cdot\tau^P)/\Gh$, and we arrive at
$$\pi_1\left(\phi^{-1}(G\cdot\tau^P)/\Gh\right)\cong\pi_1\left(\phi^{-1}(G\cdot a)/\Gh\right)$$ for
this {\em particular} $a\in\tau^P$.


For any two  values $a$ and $b$ in the same connected open face on
$\tau^P$,
 $$\pi_1\left(\phi^{-1}(G\cdot a)/\Gh\right)\cong \pi_1\left(\phi^{-1}(G\cdot b)/\Gh\right)$$
since  $\phi^{-1}(G\cdot a)$ and  $\phi^{-1}(G\cdot b)$ are
equivariantly diffeomorphic. Now, let $c\in\tau^P$ be a value on a
singular face. Take a value $a\in\tau^P$ in a connected open face
very close to $c$. Take $\bar O'\subset \tau^P$ as in
Lemma~\ref{T-remove}. Using (2) of Lemma~\ref{link} again, we can do removing in
$\phi^{-1}(G\cdot\bar O')/\Gh$ corresponding to the removing in
$\phi^{-1}(\bar O')/(T\cap \Gh)$; and we arrive at
$$\pi_1\left(\phi^{-1}(G\cdot a)/\Gh\right)\cong\pi_1\left(\phi^{-1}(G\cdot c)/\Gh\right).$$
\end{proof}


\begin{lemma}\label{remove}
 Assume the assumptions of Theorem~\ref{thm3} hold,  where $G$ is nonabelian. Let $c\in\tau$ be a value, where $\tau$ is a face of
   $\mathfrak t^*_+$ such that  $\tau\neq\tau^P$, and let $a$ be a generic value on
    $\tau^P$ very near $c$.
Let $O\subset\mathfrak{g}^*$ be a small open invariant neighborhood
of $c$ containing $a$.
    Let $B$ be the set of values in $O\cap\mathfrak t^*_+$  other than those on the open connected
    face of generic values on  $\tau^P$ containing $a$.
   Then
   $$\pi_1\left(\phi^{-1}(O)/\Gh\right)\cong\pi_1\left(\phi^{-1}(O)/\Gh-\phi^{-1}(G\cdot
   B)/\Gh\right).$$
   \end{lemma}
 \begin{proof}
Consider the cross section $R^{\tau}$ where $G_{\tau}$ acts. Note that $\tau$ lies in the central dual Lie algebra
of $G_{\tau}$.  Using Lemmas~\ref{Glink} and \ref{link}, we have
$$\pi_1\left(\phi^{-1}(O)/\Gh\right)\cong\pi_1\left(\phi^{-1}(O)/\Gh-\phi^{-1}(G\cdot \tau)/\Gh\right).$$
For other non-principal faces $\tau'$'s,  we use the cross section theorem and Lemma~\ref{link} to inductively
remove $\phi^{-1}(G\cdot\tau')/\Gh$'s. If there are singular faces on $O\cap\tau^P$,  then we use Lemmas~\ref{Tlink} and
\ref{link} to remove the rest (deforming may also be needed).
If further detail is prefered, one may refer to the proof of Lemma 6.18 in \cite{L2}.
\end{proof}



   \begin{lemma}\label{Gisom'}
     Assume the assumptions of Theorem~\ref{thm3} hold, where $G$ is nonabelian.
Let $c\in\tau$ be a value, where $\tau$ is a face of
   $\mathfrak t^*_+$ such that  $\tau\neq\tau^P$, and let $a$ be a generic value on
    $\tau^P$ very near $c$. Then
 $\pi_1\left(\phi^{-1}(G\cdot c)/\Gh\right)\cong\pi_1\left(\phi^{-1}(G\cdot a)/\Gh\right)$.
   \end{lemma}
\begin{proof}
In Lemma~\ref{remove}, we take $O$ small enough such that
$\phi^{-1}(O)$ equivariantly deformation retracts to
$\phi^{-1}(G\cdot c)$. Then
$$\pi_1\left(\phi^{-1}(O)/\Gh\right)\cong\pi_1\left(\phi^{-1}(G\cdot
c)/\Gh\right).$$ Since
$$\pi_1\left(\phi^{-1}(O)/\Gh-\phi^{-1}(G\cdot
B)/\Gh\right)\cong\pi_1\left(\phi^{-1}(G\cdot
a)/\Gh\right),$$ the claim follows from Lemma~\ref{remove}.
\end{proof}







\begin{proof} [Proof of (B) of Theorem~\ref{thm3} for nonabelian $G$] We use
Lemma~\ref{Glink}, the cross section theorem and Lemma~\ref{link} to
inductively remove $\phi^{-1}(G\cdot \tau)/\Gh$ from $M/\Gh$
for the faces $\tau$'s other than the principal face $\tau^P$ which contain values of $\phi$.
 Assume we have now
$\pi_1(M/\Gh)\cong\pi_1\left(\phi^{-1}(G\cdot\tau^P)/\Gh
\right)$. Then by Lemma~\ref{Gisom},
$\pi_1\left(\phi^{-1}(G\cdot\tau^P)/\Gh \right)\cong\pi_1\left(
\phi^{-1}(G\cdot a)/\Gh\right)$ for $a\in\tau^P$. The claim that the
$\pi_1\left( \phi^{-1}(G\cdot a)/\Gh\right)$'s are isomorphic
for all $a\in\mbox{im}(\phi)$ follows from Lemma~\ref{Gisom'}.
\end{proof}

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 \end{thebibliography}
\end{document}




0909.3179.extracted_bib
\begin{thebibliography}{99}
\bibitem{AT} M. Atiyah, {\em Convexity and commuting Hamiltonians}, Bull. Lond. Math. Soc. {\bf 14} (1982), 1-15.
\bibitem {Br} G. Bredon, {\em Introduction to Compact Transformation Groups}, Academic Press, New York, 1972.
\bibitem {GLS} V. Guillemin, E. Lerman, and S. Sternberg, {\em Symplectic fibrations and multiplicity diagrams}, Cambridge University Press, Cambridge, 1996.
\bibitem{GS0} V. Guillemin and S. Sternberg, {\em Convexity properties of the moment mapping},  Invent. Math. {\bf 67} (1982),
  no. 3, 491--513.
\bibitem {GS} V. Guillemin and S. Sternberg, {\em Symplectic techniques in physics}, Cambridge University Press, Cambridge, 1990.
\bibitem {GS1} V. Guillemin and S. Sternberg, {\em A normal form for the moment map}, Differential geometric methods in mathematical physics, (S. Sternberg, Ed.) Reidel, Dordrecht, Holland, 1984.
\bibitem{GSD} L. Godinho and M. E. Sousa-Dias, {\em The fundamental group of $S^1$-manifolds}, math.SG/0505420.
\bibitem {K} F. C. Kirwan, {\em Convexity properties of the moment mapping III}, Invent. Math. {\bf 77} (1984), 547-552.
\bibitem{K'} F. C. Kirwan,  {\em The cohomology of quotients in symplectic and algebraic geometry}, Princeton University Press, 1984.
\bibitem {Ler} E. Lerman, {\em Gradient flow of the norm squared of a moment map}, Enseign.Math. (2) 51 (2005), no. 1-2, 117-127.
\bibitem {LMTW} E. Lerman, E. Meinrenken, S. Tolman, and C. Woodward, {\em Nonabelian convexity by symplectic cuts},
                    Topology {\bf 37}, no. 2, 245--259, 1998.
 \bibitem {L1} H. Li, {\em $\pi_1$ of Hamiltonian $S^1$-manifolds}, Proc.  Amer. Math. Soc. {\bf 131}, no. 11, 3579-3582, 2003.
 \bibitem {L2} H. Li, {\em The fundamental group of symplectic manifolds with Hamiltonian Lie group actions}, Journal of Symplectic
 Geometry, Vol. 4, No. 3, 345-372, 2007.
\bibitem {M} C. M. Marle, {\em Le voisinage d'une orbite d'une action hamiltonienne d'un group de Lie}, In: S\'eminaire sud-rhodanien
de g\'eom\'etrie, $\Pi$ (Lyon, 1983), 19-35, Travaux en cours, Hermann, Paris, 1984.
\bibitem {SL} R. Sjamaar and E. Lerman, {\em Stratified symplectic spaces and reduction}, Ann. of Math. (2)  {\bf 134}, no. 2, 375--422, 1991.
\bibitem {W} C. Woodward, {\em  The Yang-Mills heat flow on the moduli space of framed bundles on a surface}, Amer. J. Math. 128 (2006), no. 2, 311-359.
 \end{thebibliography}