$\Gamma$-structures and symmetric spaces
$\Gamma$-structures are weak forms of multiplications on closed oriented manifolds. As shown by \textsc{Hopf the rational cohomology algebras of manifolds admitting $\Gamma$-structures are free over odd degree generators. We prove that this condition is also sufficient for the existence of $\Gamma$-structures on manifolds which are nilpotent in the sense of homotopy theory. This includes homogeneous spaces with connected isotropy groups. Passing to a more geometric perspective we show that on compact oriented Riemannian symmetric spaces with connected isotropy groups and free rational cohomology algebras the canonical products given by geodesic symmetries define $\Gamma$-structures. This extends work of \textsc{Albers, Frauenfelder and \textsc{Solomon on $\Gamma$-structures on Lagrangian Grassmannians.
1602.06753.tex
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\title[$\Gamma$-structures and symmetric spaces]{$\Gamma$-structures and symmetric spaces}


\author[B.\ Hanke \& P.\ Quast]{Bernhard Hanke and Peter Quast}


\address{Institut f\"{u}r Mathematik, Universit\"{a}t Augsburg, 
86135 Augsburg, Germany}
\email{hanke@math.uni-augsburg.de,\; peter.quast@math.uni-augsburg.de}


\subjclass[2010]{Primary 57T15. Secondary 53C35,  55S45, 57T25}

\keywords{$\Gamma$-structures, Postnikov decompositions, rational cohomology, symmetric spaces}





\begin{document}

\begin{abstract}
$\Gamma$-structures are  weak forms of multiplications on  closed 
oriented manifolds. As shown by \textsc{Hopf} the rational cohomology algebras of 
manifolds admitting $\Gamma$-structures are free over odd degree generators. 
We prove that  this condition is also sufficient for the existence of 
$\Gamma$-structures on manifolds which are nilpotent in the sense of homotopy 
theory. This  includes homogeneous spaces with connected isotropy groups. 

Passing to a more geometric perspective we  show that on compact oriented Riemannian symmetric spaces with 
connected isotropy groups and  
 free  rational cohomology algebras  the canonical products  
 given  by geodesic symmetries define  $\Gamma$-structures. 
This extends work of \textsc{Albers, Frauenfelder} and 
\textsc{Solomon} on  $\Gamma$-structures on Lagrangian Grassmannians.


\end{abstract}

\maketitle

\section*{Introduction} In his seminal papers \cite{Ho1, Ho2} on the (co)homological structure 
of Lie groups \textsc{Hopf}  introduced the notion 
of {\em $\Gamma$-manifolds}. By definition these are closed connected oriented manifolds 
$M$ together with continuous maps 
\[
   \psi : M \times M \to M 
\]
so that the mapping degrees of the two restrictions $\psi_{p}: y \mapsto \psi(p,y)$ and 
$\psi^q: x \mapsto \psi(x,q)$ 
are non-zero for some (and hence for all) $p, q \in M$. In some sense such maps  $\psi$, which we 
call  {\em $\Gamma$-structures}, capture the simplest non-trivial homological information of  
Lie group multiplications. 
\textsc{Hopf} proved that the rational cohomology rings of $\Gamma$-manifolds are of a surprisingly restricted 
type: Because they admit compatible  comultiplications (and are hence  {\em Hopf algebras} in modern terminology) 
they are free graded commutative $\Q$-algebras, whose generators must be in odd 
degrees as $M$ is finite dimensional, 
 see \cite{Ho1, Ho2} and \cite[Chap.\ VI, \S 2.A]{D} for further information.
As
carried out by \textsc{Borel} in \cite[Chapitre II, $\S$ 7]{Bo}, further divisibility restrictions on the mapping degrees of $\psi_p$ 
and $\psi^q$ have similar  implications for the cohomology rings over finite fields. 
%We learned from \textsc{Dieter Kotschick} that spaces with $\Gamma$-structures 
%have virtually abelian fundamental groups (see Lemma \ref{LEM: virtually abelian}).
 \par

A closely related and much better known structure is that of an  {\em H-space}, with both restrictions 
$\psi_p$ and $\psi^q$ being homotopic to the identity. 
While every compact connected oriented manifold that is an 
H-space is obviously a $\Gamma$-manifold, the converse fails: All odd dimensional 
unit spheres are
 $\Gamma$-manifolds (see \cite{Ho2} and our Theorem \ref{THM: Rank split} for a generalization), 
 but \textsc{Adams} celebrated 
 `Hopf invariant one theorem' in \cite{Ad}
 says that only spheres of dimension $1$, $3$ or $7$ admit
 H-space structures.
  
 In the first part of our paper, Section \ref{SEC:Homotopy}, we shall make some general remarks on the existence of $\Gamma$-structures on manifolds satisfying the above cohomological condition. As the only non-trivial requirement 
 for a $\Gamma$-structure $\psi : M \times M \to M$ 
 is the non-vanishing of the mapping degrees of $\psi_p$ and $\psi^q$,  a construction of such structures 
 by obstruction theory requires the separation of rational and torsion information in the homotopy 
 type of $M$. This is the underlying idea of {\em rational homotopy theory}, which works 
 best for spaces whose Postnikov decompositions consist of principal fibrations and can hence 
 be described by accessible cohomological invariants. Examples are   {\em simple} spaces, 
 whose fundamental groups are abelian and  act trivially on higher homotopy groups, and, more
 generally, {\em nilpotent} spaces, whose  fundamental groups are nilpotent and 
   act nilpotently on higher homotopy 
groups. 

In Section \ref{SEC:Homotopy} we will prove the following 
converse of \textsc{Hopf's} result. From now on the notion {\em free algebra} stands for {\em free graded commutative 
algebra over the rationals}. 

\begin{thm} \label{existence}  Let $M$ be a closed connected oriented manifold which is 
nilpotent as a topological space. If  $H^*(M; \Q)$ is a free algebra -
necessarily over odd degree generators - then $M$ admits a $\Gamma$-structure. 
\end{thm} 

This is in sharp contrast to the existence of H-space structures, which is a much more restrictive property. 

\begin{cor} \label{homogenous} 
Let $M = G/H$ be a compact connected homogeneous space where  $G$ is a Lie group and $H < G$ is a 
closed connected subgroup. 
If $H^*(M;\Q)$ is a free algebra, then $M$ admits a $\Gamma$-structure. 
\end{cor} 

Theorem \ref{existence} also implies (see Corollary \ref{free_virtab}) that nilpotent manifolds with 
free rational cohomology algebras have  virtually abelian fundamental groups. 


Our abstract 
existence results motivate the search for explicit geometric constructions of $\Gamma$-structures. 
Already \textsc{Hopf} \cite{Ho2} used geodesic symmetries on odd dimensional spheres 
to write down $\Gamma$-structures. 
It is therefore natural to consider Riemannian symmetric spaces $P$
endowed with their  \emph{canonical products}
(see e.g.\ \cite[Chap.\ II, \S 1]{L-I})
 \begin{equation}
 \label{EQ: Product sym space}
\Theta:P\times P\to P,\quad (x,y)\mapsto s_x(y),
 \end{equation}
 where $s_x$ denotes the \emph{geodesic symmetry} of $P$ at the point $x\in P,$ that is
the involutive isometry of $P$ that fixes $x$ and that reverses 
 the direction of all geodesics emanating from $x.$
 In Section \ref{SEC: can prod sym space} we investigate in which cases
 the canonical product $\Theta$ of a compact oriented symmetric space
 $P$ is a $\Gamma$-structure. This
leads to a  proof of the following conclusive result.


\begin{thm}
 \label{THM: classification} 
 Let $P$ be a  compact symmetric space and 
 let $G$ be the identity component of its isometry group.
 Assume that the isotropy subgroup 
 $K\subset G$ of a base point in $P$ 
 is connected. 
Then the following assertions 
are equivalent:
\begin{itemize}
\item $H^*(P;\Q)$ is a free algebra.
\item The canonical product on $P$ is a $\Gamma$-structure.
\end{itemize}
\end{thm}

We remark that whenever  the canonical product $\Theta$ defines  a 
$\Gamma$-structure on $P$, then the mapping degrees of 
$\Theta_p$ and $\Theta^q$ are (up to sign)  powers of $2$ so that by  \cite{Bo} additional restrictions 
on the cohomology algebras over finite fields of characteristic different from 
two are implied ex post by properties of the rational cohomology algebras.\par

Unfortunately Theorem \ref{THM: classification} does not cover all the manifolds from Corollary \ref{homogenous}. For 
example 
it remains an open problem to provide a geometric construction of $\Gamma$-structures on complex 
and quaternionic Stiefel manifolds (see \cite[Thm.\ 3.10, p.\ 119]{MT}), which are not symmetric spaces. 



\subsection*{Acknowledgements}
We are most grateful to \textsc{Jost-Hinrich Eschenburg, Urs Frau\-en\-felder, Oliver Goer\-tsches, 
Dieter Kot\-schick, Mar\-kus Up\-meier} and \textsc{Mi\-cha\-el Wie\-meler}
for valuable remarks 
during the preparation of this manuscript. The research of the first named author was supported 
by DFG grant HA 3160/6-1.

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\section{Postnikov decompositions and $\Gamma$-structures} 
\label{SEC:Homotopy} 

In this section we present  a  homotopy theoretic construction of $\Gamma$-structures.
Recall  that the homotopy type of a  path connected CW complex $X$
can be analysed by means of its {\em Postnikov decomposition}, see, for example,  \cite[p. 410 ff.]{Ha1}: 
Let $X_n$ be obtained by killing 
all homotopy groups of $X$ above degree $n$ by attaching cells of dimension at least $n+2$.
Up to homotopy equivalence we can assume that each inclusion $p_{n+1}: X_{n+1} \to X_n$ is a fibration. 
The long exact homotopy sequence shows that the fibre is an Eilenberg--MacLane 
space $K(\pi_{n+1}, n+1)$ where $\pi_{n+1} := \pi_{n+1}(X)$. In particular, $X_1$ is 
 the classifying space $B\pi_1(X)$. 

Recall that $X$  is called {\em simple}, if $\pi_1(X)$ is 
 abelian and acts trivially on higher homotopy groups. 
For simple $X$ each fibration 
$p_{n+1}: X_{n+1} \to X_n$  is {\em principal}, see \cite[Theorem 4.69]{Ha1},  that is
the pull back of the path-loop fibration 
\[
   K(\pi_{n+1}, n+1) \to PK(\pi_{n+1} , n+2) \to K(\pi_{n+1}, n+2)
 \]
along a map $X_{n} \to K(\pi_{n+1}, n+2)$ which defines  the $n$-th {\em $k$-invariant}  
$k_n \in H^{n+2}(X_{n}; \pi_{n+1})$.
This class is equal to the image of the fundamental class in $H^{n+1}(K(\pi_{n+1} , n+1) ; \pi_{n+1})$ 
under the transgressive differential $d_{n+2}$ in the Leray--Serre spectral sequence for the fibration $p_{n+1}$. 
We denote by $(k_n)_{\Q} \in H^{n+2}(X_{n}; \pi_{n+1} \otimes \Q)$ the image of $k_n$ under 
the coefficient homomorphism $\pi_{n+1} \to \pi_{n+1} \otimes \Q$. 

In the following we collect some well known facts on the cohomology of Eilenberg--MacLane 
spaces. 

\begin{lem} \label{computation} 
Let $C$ be a cyclic group, and let $m, n >0$ be positive natural numbers. Then
\begin{itemize} 
     \item $H^*(K(C,n) ; \Z)$ is a finitely generated group in each degree. 
    \item for $C = \Z$ the cohomology algebra $H^*(K(\Z,n) ; \Q)$ is free with one generator in degree $n$. 
   \item for $|C| < \infty$ the reduced cohomology $\widetilde{H}^*(K(C,n); \Q)$ is equal to $0$.
  \end{itemize} 
Let $\mu_m : C \to C$ be the multiplication by $m$. Then
\begin{itemize}
   \item for $C = \Z$  the induced map 
\[
      \mu_m^* : H^*(K(\Z,n) ; \Q) \to H^*(K(\Z,n) ; \Q) 
\]
is given by multiplication with $m^k$ on $H^{kn}(K(\Z,n) ; \Q)$. 
\item for all $C$ the map 
\[
   \mu_m^* : \widetilde{ H}^* (K(C,n) ; \Z/m ) \to \widetilde{H}^*(K(C,n) ; \Z/m)
\]
is equal to $0$. 
\end{itemize} 
\end{lem}  

\begin{proof} Apart from the  last  assertion the lemma can be proven fairly directly by induction on $n$. 
For $n=1$ we have $K(C,1) = BC$, the classifying space of $C$, so that the assertions are clear 
for $C = \Z$ (as  $B\Z = S^1$).
For $|C| < \infty$ the classifying space $BC$ is an infinite dimensional lens space, the cohomology 
$\widetilde{H}^*(BC; \Z)$ is equal to $\Z / |C|$ in even degrees and $0$ in odd degrees, and $\mu^*_m : 
H^*(BC ; \Z) \to H^*(BC; \Z)$ is 
given by multiplication with $m^k$ on $H^{2k}(BC ; \Z)$. Together with the universal coefficient 
theorem this implies all the assertions of the lemma for $n=1$. 

For the induction step we recall  that  the cohomology with coefficients in some commutative ring 
$R$ of base and fibre of the path 
loop fibration 
\[
     K(C, n) \to PK(C, n+1) \to K(C,n+1) 
\]
appear on the two coordinate axes of the $E_2$-term of the Leray-Serre spectral sequence and 
that this spectral sequence is natural with respect to homomorphisms $C \to C$. 
The spectral sequence converges to $H^*(PK(C, n+1); R)$, which vanishes in positive degrees, because $PK(C,n+1)$ is contractible. 

By induction this shows that $H^*(K(C, n+1); \Z)$ is 
finitely generated in each degree, that $\widetilde{H}^*(K(C, n+1); \Q) = 0$ for $|C| < \infty$, and  that $H^*(K(\Z,n+1); \Q)$ is a free algebra in one generator of degree $n+1$.  The last implication uses 
the multiplicative structure of the spectral sequence. 

Now let $m > 0$ be a natural number and $\mu_m : \Z \to \Z$ be multiplication by $m$. Then the naturality of the spectral sequence shows inductively 
that $\mu_m^* : H^{*}(K(\Z, n+1); \Q) \to H^*(K(\Z, n+1); \Q)$ is multiplication 
by $m^k$ in degree $k(n+1)$. 

We are left with showing that for all $C$ and all $n$ the map 
\[
   \mu_m^* : \widetilde{ H}^* (K(C,n) ; \Z/m ) \to \widetilde{H}^*(K(C,n) ; \Z/m)
\]
is equal to $0$.  First, let us assume 
that $m = p$ is a prime number. As $\widetilde{H}^*(K(C,n); \Z/p) = 0$, 
if the order of $C$ is finite and coprime to $p$, (this fact can be shown by induction on $n$), 
we can restrict attention to the 
case $C = \Z$ or $C = p^r$. It is well known, 
see \cite{Ca, Se}, or \cite[Theorem 6.19]{McCl}, 
that $H^*(K(C,n); \Z/p)$ is a polynomial algebra with free  generators of the form 
$\mathcal{P}(\iota_n)$ where $\iota_n \in H^n(K(C,n);\Z/p)$ is the fundamental class  and 
$\mathcal{P}$ is some mod $ p$ cohomology operation. Because the map $\mu_p^*$ is zero 
on $H^n(K(C,n); \Z/p)$ the assertion is hence implied by the naturality of the operations $\mathcal{P}$. 

Next, for arbitrary cyclic $C$ and $m = p^r$ with a prime number $p$, 
the assertion follows  by induction on $r$ by use of the exact Bockstein sequence 
\[
    \cdots \rightarrow \widetilde{H}^*(K(C,n); \Z/p^{r-1}) \rightarrow \widetilde{H}^*(K(C,n); \Z/p^{r}) \to 
    \widetilde{H}^*(K(C,n); \Z/p) \rightarrow \ldots , 
\]
which is natural with respect to homomorphisms $C \to C$, observing that $\mu_{m} = \mu_{p^{r-1}} \circ 
\mu_p$. 


Finally, for $m = p_1^{r_1} \cdot \ldots \cdot p_k^{r_k}$ with pairwise different primes  $p_i$, 
the map $\mu_m^*$ is zero on 
$\widetilde{H}^*(K(C, n); \Z/m)$ by splitting this cohomology according to the primary
decomposition $\Z/m \cong \Z / p_1^{r_1} \times  \cdots \times \Z /  p^{r_k}$.


 \end{proof} 

Let $X$ be  a connected finite simple CW complex whose rational 
cohomology  algebra is free. It must be finitely generated with all generators in odd degrees, because 
$X$ is assumed to be finite. Serre's finiteness theorem (or a direct
inspection of the Postnikov decomposition in connection with Lemma \ref{computation}) 
implies that the homotopy groups 
of $X$ are finitely generated in each degree. Hence they are finite products of cyclic groups. 

\begin{lem} \label{torsion} For each $n >0 $ the following holds. 
\begin{itemize} 
    \item $H^*(X_n; \Q)$ is a free algebra with generators in degrees $\leq n$ corresponding
     to  the duals of the generators of $\pi_{i}(X) \otimes \Q$, $i \leq n$.
    \item The canonical map $X \to X_n$ induces an injective map in rational cohomology. 
    \item The rationalized $k$-invariant $(k_n)_{\Q}  \in H^{n+2}(X_{n} ; \pi_{n+1} \otimes \Q)$ vanishes. 
\end{itemize} 
 \end{lem} 

 \begin{proof} The first assertion implies the second one, because 
 the canonical map $X \to X_n$ induces an isomorphism in rational cohomology up to degree $n$
 and the cohomology algebra $H^*(X; \Q)$ is free. The third
 assertion is then also  immediate because $(k_n)_{\Q}$ is the image of the fundamental class 
 in $H^{n+1}(K(\pi_{n+1} , n+1) ; \Q)$ under the differential $d_{n+2}$ and the map $X \to X_n$ 
 factors through $X_{n+1}$ so that $H^*(X_{n}; \Q) \to H^*(X_{n+1};\Q)$ is also injective. 

 It is hence enough to prove the first assertion by induction on $n$. 
 
The first assertion is clear for $n=1$ by Lemma \ref{computation} and the K\"unneth theorem.  In the inductive step 
the assumption $(k_n)_{\Q} = 0$ implies 
 \[
    H^*(X_{n+1} ; \Q) \cong H^*(X_{n}; \Q ) \otimes H^*(K(\pi_{n+1}, n+1); \Q). 
\]
Hence the assertion is again implied by Lemma \ref{computation} and the K\"unneth theorem. 
\end{proof} 

\begin{prop} \label{multi} Let $n>0$. Then for any $m > 0$  there is a self map $f _n: X_n \to X_n$ with the following properties. 
\begin{itemize} 
  \item The induced map $f_n^* : H^*(X_n; \Q) \to H^*(X_n;\Q)$ is an isomorphism. 
  \item The induced map $f_n^* : \widetilde{H}^*(X_n ; \Z/m) \to \widetilde{H}^*(X_n; \Z/m)$ is equal to $0$. 
 \end{itemize} 
 \end{prop}
 
 \begin{proof} Induction on $n$, the  assertion being immediate for $n=1$ by Lemma \ref{computation} and 
 the K\"unneth theorem. 
 By Lemma \ref{torsion} 
the $k$-invariant $k_n \in H^{n+2}(X_n ; \pi_{n+1})$ is a 
torsion class of order $\kappa$, say.  This has two useful implications.  First, by the inductive hypothesis
we find an $f_n : X_n \to X_n$ inducing an isomorphism in rational cohomology and
satisfying $f_n^*(k_n) = 0$. This results in a commutative diagram 
\[
  \xymatrix{  K(\pi_{n+1}, n+1) \ar[d]^{\rm incl.} \ar[r]^{=}& K(\pi_{n+1}, n+1) \ar[d] \\
      X_{n} \times K(\pi_{n+1}, n+1)\ar[r]^-{\alpha}  \ar[d]^{\rm proj.} &         X_{n+1} \ar[d]^{p_{n+1}} \\ 
                                   X_n      \ar[r]^{f_{n} }    &         X_n          }
 \]
Second, we claim that multiplication $h : \pi_{n+1} \to \pi_{n+1}$ with $\kappa$  gives rise to  a commutative diagram 
\[ 
  \xymatrix{   K(\pi_{n+1} , n+1) \ar[r]^{K(h, n+1)} \ar[d] & K(\pi_{n+1}, n+1) \ar[d]^{\rm incl.} \\
    X_{n+1}                           \ar[r]^-{\beta} \ar[d]^{p_{n+1}} &   X_{n} \times K(\pi_{n+1}, n+1)\ar[d]^{\rm proj.} \\ 
                                          X_n                                                     \ar[r]^{=}    &         X_n           }
 \]
 The only non-trivial task is the construction of the map $X_{n+1} \to K(\pi_{n+1}, n+1)$ appearing in the middle 
 horizontal line. For this we consider the diagram 
\[ 
  \xymatrix{   K(\pi_{n+1} , n+1) \ar[r]^{=} \ar[d] & K(\pi_{n+1}, n+1)  \ar[rr]^-{K(h, n+1)} \ar[d] & & K(\pi_{n+1}, n+1) \ar[d] \\
    X_{n+1}                           \ar[r]  \ar[d]  &   PK(\pi_{n+1} , n+2) \ar[rr]^-{PK(h, n+1)} \ar[d] & & PK(\pi_{n+1}, n+2) \ar[d] \\
        X_n                              \ar[r]^-{k_n}    &       K(\pi_{n+1} , n+2)   \ar[rr]^-{K(h, n+2)} &   &    K(\pi_{n+1} , n+2)     }
 \]
By assumption the composition in the lower row is homotopic to a constant map. Hence, by 
the homotopy lifting property, the composition
$X_{n+1} \to PK(\pi_{n+1}, n+2)$ in the second row  is fibrewise homotopic to a map into just one fibre 
$K(\pi_{n+1}, n+1)$ of the path loop fibration appearing in the right column. This
concludes the construction of  the map  $X_{n+1} \to K(\pi_{n+1}, n+1)$. 

 
 Note that the maps $\alpha$ and $\beta$ appearing in the first two diagrams 
 induce isomorphisms in rational cohomology by a 
spectral sequence argument. 

Now let $m > 0$ be arbitrary. Using the induction hypothesis, Lemma \ref{computation} and the K\"unneth formula 
it is easy to construct a self map 
\[
 f':   X_{n} \times K(\pi_{n+1}, n+1) \to X_{n} \times K(\pi_{n+1}, n+1)
\]
with the required properties. But then the composition 
\[
  f_{n+1} : X_{n+1} \stackrel{\beta}{\longrightarrow} X_{n} \times K(\pi_{n+1}, n+1) \stackrel{f'}{\longrightarrow} X_{n} \times K(\pi_{n+1}, n+1) 
  \stackrel{\alpha}{\longrightarrow}  X_{n+1}
\]
is as required. 
\end{proof}

\begin{prop} \label{exist_gamma} Let $X$ be a connected 
finite CW-complex which is a simple topological space and whose 
rational cohomology algebra  is free. Then there is 
a continuous map 
 \[
   \psi : X \times X \to X 
\]
 so that for all $p,q  \in X$ the maps $\psi(p, -): X \to X$ and $\psi(-,q): X \to X $ induce isomorphisms in rational cohomology. 
\end{prop} 

\begin{proof} It is clear that such a map 
exists on $X_1$. Assume that we have already constructed a map $\psi_n : X_n \times X_n \to X_n$ 
with the required properties. 
 
Taking the existing product on $X_n$ and the product on $K(\pi_{n+1} , n+1)$
induced by the addition map on $\pi_{n+1}$ we obtain a multiplication $\psi'$ 
on $X' := X_n \times K(\pi_{n+1},n+1)$ 
with the required properties. Now we define $\psi_{n+1}$ as the composition 
\[  
  X_{n+1} \times X_{n+1} \stackrel{\beta \times \beta}{\longrightarrow}  X' \times X' 
  \stackrel{\psi'}{\longrightarrow} X' \stackrel{\alpha}{\longrightarrow} X_{n+1} .
\]
where $\alpha$ and $\beta$ are taken from the proof of Proposition \ref{multi}. Because these maps induce
isomorphisms in rational cohomology, the map $\psi_{n+1}$ has the required properties. 

Once we have constructed a map $\psi_{n}$ with $n > 2 \dim X$, the construction of $\psi$ is complete by 
the cellular approximation theorem. 
\end{proof} 

For the next notions and results compare \cite[Sections 3.1. and 3.2]{MP}. 
A path connected CW complex  $X$ is called {\em nilpotent}, if $\pi_1(M)$ is 
a nilpotent group and acts nilpotently on the higher homotopy groups. 
Note that simple complexes are automatically nilpotent. 
For a nilpotent complex the fibrations $p_{n+1}: X_{n+1} \to X_n$ in the Postnikov decomposition 
are in general not principal, but they admit 
{\em finite principal refinements}
\[
      X_{n+1} =: Y_{r_n} \stackrel{q_{r_n}}{\longrightarrow} Y_{r_n - 1} \stackrel{q_{r_n-1}}{\longrightarrow} \ldots Y_1 \stackrel{q_1}{\longrightarrow} Y_0 := X_n,
 \]
 where each $q_i : Y_i \to Y_{i-1}$ is a principal fibration with fibre $K(G_i, n+1)$ for some abelian group $G_i$, which 
 is classified by a cohomology class in $H^{n+2}(Y_{i-1} ; G_i)$. The same argument as in the proof 
 of Proposition \ref{exist_gamma} then shows: 
 
 
 
 
 \begin{prop} \label{exist_nilpotent} Let $X$ be a connected finite CW complex which is a nilpotent topological space. Assume that 
 the rational cohomology algebra of $X$ is free. Then there is a map 
 $\psi: X \times X \to X$ so that  for all $p,q  \in X$ the maps $\psi(p,-) : X \to X$ and $\psi(-,q): X \to X$ 
 induce isomorphisms in rational cohomology. 
 \end{prop} 

From this Theorem \ref{existence} follows. 

\begin{lem} \label{simple_space}  Let $M = G / H$ be a homogeneous space where $G$ is a connected Lie group and $H < G$ 
be a closed connected subgroup. Then $M$ is a simple topological space.
\end{lem} 

\begin{proof} This fact is well known and we include a proof for the readers' convenience. 
Because $H$ is connected we get an exact sequence
\[
    \ldots \to \pi_2(G/H) \to \pi_1(H) \to \pi_1(G) \to \pi_1(G/H) \to 1.
\]
The topological group $G$ has abelian fundamental group, and  hence the same holds for $G/H$. 

Next, let $f : S^n \to G/H$ and $\mu : S^1 \to G/H$ be based maps, where we take the south pole of 
any sphere as base point $*$.
Recall that $[\mu]_*([f]) \in \pi_n(G/H)$, the result of the action of $[\mu] \in \pi_1(G/H; *)$ on $[f]\in \pi_n(G/H; *)$, 
 is represented by the following map $\mu *  f : S^n \to G/ H$. Consider the one point union 
$S^n \vee S^n$ where the second sphere is piled above the first one, identifying the north pole of the 
first with the south pole of the second. We take the south pole of the first sphere as base point of $S^n \vee S^n$. 
Now consider the composition $S^n \stackrel{\text{height}}{\longrightarrow}  [0,1] \stackrel{\mu}{\rightarrow} G/H$ on the left hand sphere, and the map $f : S^n \to G/H$ on the right hand sphere, and compose 
the resulting map $S^n \vee S^n \to G/H$ 
with the base point preserving coproduct $S^n \to S^n \vee S^n$. This defines $\mu * f$. 
Note that for $n=1$ this results in the usual conjugation action of $\pi_1(G/H;*)$ on itself. 

By  the above exact sequence the map $\mu$ lifts to a based
map $\overline{\mu} : S^1 \to G$. Using the left multiplication  of $G$ on $G/H$ and 
the above explicit description of $\mu * f$ 
it is easy to show that $\mu * f$ is based homotopic to $f$. \end{proof}

Together with our previous results this implies Corollary \ref{homogenous}.  

%The structure results of rational graded Hopf algebras also cover algebras with generators in arbitrary large 
%degrees: By  \cite{Ho2} they are isomorphic to tensor products of a free algebra with generators in even 
%degrees and an exterior algebra with generators in odd degrees. Note that in the cohomology rings 
%of (finite dimensional) manifolds no even degree generator can occur. But if we allow 
%infinite dimensional spaces Proposition \label{exist_nilpotent} can be generalized as follows. 

%\begin{prop} Let $X$ be a path connected nilpotent CW complex with finite skeleta. Assume that $H^*(X; \Q)$ 
%is a free algebra (hence a product of a polynomial algebra and an exterior algebra). Then there is a product map 
%\[  
%     \psi : X \times X \to X 
%\]
%so that for each $x \in X$ the induced maps on $\psi(x, -)$ and $\psi(-,x)$ are injective in rational homology. 
%\end{prop} 

%The proof is similar to the proof of Proposition \ref{exist_nilpotent} above, with the only exception that 
%the induction cannot be finished after finitely many steps in the Postnikov decomposition. 

% is in accordance with the fact \cite{Ho2} that rational graded Hopf algebras which are finitely generated in 
%each degree are free graded algebras with finitely many generators in each degree. Because this 
%general fact is not important for the main results of our paper we omit a detailed proof at this point. 


\begin{rem} Of course the above argument is modelled along the lines of 
rational homotopy theory. In particular our Proposition \ref{multi} is implied
by \cite[Theorem (12.2)]{Sul} (note that by Lemma \ref{torsion} our $X$ is a formal space 
in the sense of rational homotopy theory). However, a complete proof of Sullivan's theorem 
has been carried out in the literature only for simply connected spaces \cite{Sh} by 
obstruction theory based on cellular decompositions. Because finite skeleta of 
non-simply connected nilpotent spaces are in general not nilpotent \cite{Le}, 
the non-simply connected case requires a different approach based on Postnikov decompositions
(as indicated already in \cite{Sul}). This issue and the special focus of our paper let
us prefer the  somewhat ad hoc, but self-contained discussion above  over an in-depth exploration 
of rational homotopy theory. 
\end{rem} 

We are grateful to \textsc{Dieter Kotschick} for pointing out the following lemma and corollary. 


\begin{lem}\label{LEM: virtually abelian} Each $\Gamma$-manifold has virtually abelian fundamental group. 
\end{lem} 

\begin{proof} Let $\psi: M \times M \to M$ be a $\Gamma$-structure. Without loss of generality
we can pick points $p, q \in M$ so that $\psi_p, \psi^q  : M \to M$ preserve a base point in $M$. 
Let $H_1, H_2 < \pi_1(M)$ be defined as the images of 
\[
      (\psi_p)_* : \pi_1(M) \to \pi_1(M) \; ,\quad  (\psi^q)_* : \pi_1(M) \to \pi_1(M) . 
\]
The map  $\psi_{p}, \psi^q : M \to M$ factor through the connected coverings of $M$ 
defined by $H_1$ and $H_2$, respectively. Because $\psi_p$ and $\psi^q$ have non-zero 
mapping degrees, these coverings are finite and hence $H_1$ and $H_2$ are of finite index in $\pi_1(M)$. 
This implies that also $H_1 \cap H_2 < \pi_1(M)$ is of finite index. 

The map $\psi_* : \pi_1(M) \times \pi_1(M) \to \pi_1(M)$ being a group homomorphisms, elements 
in $H_1$ commute with elements in $H_2$. This implies that the finite index 
subgroup $H_1 \cap H_2 < \pi_1(M)$ is 
abelian. 
\end{proof} 

Together with Theorem \ref{existence} this implies

\begin{cor} \label{free_virtab} Let $M$ be a closed connected oriented manifold which is 
nilpotent as a topological space. If  $H^*(M; \Q)$ is a free algebra -
necessarily over odd degree generators - then $\pi_1(M)$ is virtually abelian. 
\end{cor} 

We do not know whether this conclusion can be drawn without  applying Theorem \ref{existence}. 



 


\section{Canonical products on  symmetric spaces}
\label{SEC: can prod sym space}
In this section we study in which cases the canonical product $\Theta$  on a compact oriented symmetric space
 $P$   defined in  (\ref{EQ: Product sym space}) is a $\Gamma$-structure. 
First we observe that the degree of the map
 $\Theta_p:P\to P,\; y\mapsto \Theta(p,y)=s_p(y)$
with $p\in P$ fixed is $$\det(\Theta_p)=(-1)^{\dim(P)}.$$ \par
For fixed $q\in P$ we will examine  the map
 \begin{equation}
\label{EQ: theta}
 \theta:=\Theta^q:P\to P,\quad x\mapsto  \Theta(x,q)=s_x(q).
\end{equation}
If $\deg(\theta)\neq 0,$ 
$\Theta$ is a $\Gamma$-structure. 
We use of some features of compact symmetric spaces which can be found in the classical literature such as  \cite{He,L-I, L-II}
or \cite[Chap.\ 8]{W}.


\subsection{The mapping degree of $\theta$}
Our key observation is that
$\theta$ 
is the \emph{squaring map} w.r.t.\ the origin $q$
in the sense of \textsc{Loos} \cite[Chap.\ II, \S1]{L-I}.
Indeed,  let $\gamma$ be a geodesic in $P$ with $\gamma(0)=q,$ then 
$\theta(\gamma(1))=s_{\gamma(1)}(\gamma(0))=\gamma(2).$ 
For short
\begin{equation}
 \label{EQ: theta as dilation}
 \theta\circ \Exp_q=\Exp_q\circ \mu_2,
\end{equation}
where $\mu_2:T_qP\to T_qP$ denotes the multiplication by the scalar 2 and 
$\Exp_q$ the Riemannian exponential map of $P$ at $q.$
Now the pre-image of a point $y\in P$ is 
\begin{equation}
 \label{EQ: Urbild}
 \theta^{-1} (\{y\})
= \{x\in P\; :\; x\; \text{is the midpoint of a geodesic arc in $P$ between $q$ and $y$}\}.
\end{equation}
Differentiating Equation (\ref{EQ: theta as dilation}) at a non-zero point $X\in T_qP$ 
yields
\begin{equation}
 \label{EQ: differential theta 1}
 d_{\Exp_q(X)} \theta\big( J(1)\big)=J(2),
\end{equation}
whenever $J$ is a Jacobi field with $J(0)=0$ along the geodesic 
$t\mapsto \Exp_q(tX).$ \par
 
To fix the notations and for the sake of completeness we first determine  some
Jacobi fields on $P$ explicitly. Our approach is standard. 
For convenience we fix a \emph{flat} $A\subset P,$ 
that is a maximal dimensional 
connected totally geodesic submanifold of $P$ whose  sectional 
curvature vanishes,  with $q\in A.$ Flats in compact symmetric spaces 
are embedded flat tori.
Since the curvature tensor $R$ of $P$  is parallel and $A$ is flat, the self-adjoint
operators 
$$T_qP\to T_qP,\; \quad 
V\mapsto R(V,X)X,$$
with $X\in T_qA$ commute. 
Thus, if $P\neq A,$ there exists a collection 
$$\Spec:=\{\lambda_1,\dots, \lambda_s\}$$
of distinct quadratic forms $\lambda_j:T_qA\to\R$ (the squares of the roots of $P$) 
so that the spaces 
$$E_{\lambda_j}:=\big\{V\in T_qP\; :\; R(V,X)X=\lambda_j(X)\cdot V\; \text{for all}\; X\in T_qA\big\},
\quad j=1,2,\dots, s,$$
have positive dimension, and 
$T_qP$ is the orthogonal direct sum 
\begin{equation}
 \label{EQ: Decomposition of TP}
T_qP=T_qA\oplus E_{\lambda_1}\oplus E_{\lambda_2}\oplus \dots \oplus E_{\lambda_s}.
\end{equation}
Since $A$ is a  flat, 
no $\lambda\in\Spec$ vanishes identically. 
Moreover $\lambda\geq 0$ for all $\lambda\in\Spec,$
because $P$ has non-negative sectional curvature.
Recall that a Jacobi field $J$ along a non-constant geodesic $\gamma$ in $P$ 
is a solution of $\nabla_{\dot{\gamma}}\big(\nabla_{\dot{\gamma}} J\big)+R(J,\dot{\gamma})\dot{\gamma}=0,$
 where $\nabla$ is the Levi-Civita connection  of $P.$ 
As $R$ is parallel, the Jacobi field $J$  with $J(0)=0$ and $(\nabla_{\dot{\gamma}}J )(0)=V\in T_qP$ 
along a non-constant geodesic
$\gamma$ in $A$ starting in $q$ satisfies
\begin{equation}
\label{EQ: Jacobi fields 1}
 J(t)=\left\{\begin{array}{cl}
               \frac{1}{\sqrt{\lambda(\dot{\gamma}(0))}}\sin\big(t\sqrt{\lambda(\dot{\gamma}(0))}\big)
               \cdot V^{\|}(t), & \text{if} \;
               V\in E_{\lambda},\; \lambda\in\Spec,\;  \lambda(\dot{\gamma}(0))\neq 0\\
               t\cdot V^{\|}(t) ,&\text{if} \;
              V\in T_qA\; \text{or}\;V\in E_{\lambda},\;  \lambda(\dot{\gamma}(0))= 0 
              \end{array}\right.,
\end{equation}
where $V^{\|}$ denotes the parallel vector field along $\gamma$ with 
$V^{\|}(0)=V.$ \par

We are now able to find a regular value of $\theta.$
A point $a\in A$ is called \emph{regular} 
w.r.t.\  $q$, if $a$ is not an element of the cut locus of $P$ w.r.t.\ $q$
and if $A$ is the unique flat in $P$ that contains both points $a$ and $q.$
Otherwise  $a$ is  called \emph{singular} w.r.t.\ $q.$
The set $\Sing$ of all singular points in $A$ w.r.t.\ $q$ 
is a closed subset of measure zero in $A$ (see e.g.\ \cite[Chap.\ VI]{L-II}).
For $A$ is totally geodesic, $\theta$ leaves $A$ invariant. Since 
$\theta$ is  smooth, 
$\theta(\Sing)$ is again a compact null set of $A.$
We take a point
$$y\in
A\setminus (\Sing\cup\theta(\Sing)).$$
We will see that (up to a slight modification) $y$ is a regular value of $\theta.$
As every geodesic in $P$ lies in some flat
and $A$ is the only flat containing $q$ and $y,$ 
we get by Equation  (\ref{EQ: Urbild})
\begin{equation}
\label{EQ: Urbild flat}
 \theta^{-1} (\{y\})
 =\{x\in A\; :\; x\; \text{is the midpoint of a geodesic arc in $A$ between $q$ and $y$}\}.
\end{equation}
Let  $\gamma$ be
the shortest geodesic in $P$ with $\gamma(0)=q$ and 
$\gamma(1)=y$  and set $Y:=\dot{\gamma}(0).$
The restriction of the Riemannian exponential to $T_qA$ 
is a covering of $A.$
By Equation 
(\ref{EQ: theta as dilation}) the lift
of $\theta|_A$ to $T_qA$ is the multiplication by 2. So we get
$$
 \theta^{-1}(\{y\})
= \Exp_q\left(\tfrac{1}{2}Y+\tfrac{1}{2}\LambdaL\right)\quad
\text{with}\quad\LambdaL:=\{X\in T_qA\; :\; \Exp_q(X)=q\}.$$
Let $r=\mathrm{rank}(P):=\dim(A)$
be the \emph{rank} of $P.$ Taking a basis $\{B_1,\dots, B_r\}$ of $T_qA$ 
that generates $\LambdaL,$ i.\ e.\ 
$\LambdaL=\mathrm{span}_{\Z}(B_1,\dots, B_r),$ we get
  $$\theta^{-1}(\{y\})=\Big\{\Exp_q\Big(\tfrac{1}{2}Y+\tfrac{1}{2}\sum_{j=1}^r\delta_jB_j\Big)\; :\; 
  \delta_j \in \{0,1\}\;\; \text{for}\;\; j=1,\dots, r\Big\}.$$
 Since $\Exp_q$ is injective on $\Big\{ 
 \frac{1}{2}Y+\frac{1}{2}\sum\limits_{j=1}^r\delta_jB_j:\;  \delta_j\in\{0,1\}\; \text{for}\; j=1,\dots, r\Big\},$
 the pre-image   $\theta^{-1}(\{y\})$  has cardinality  $2^{r}$ 
 (cf.\ {\cite[p.\ 933]{AFS}}). 
We set $$\theta^{-1}(\{y\})=\{x_1,\dots,x_{2^r}\}.$$\par
Since $y\notin \Sing\cup\theta(\Sing),$ each point in $\theta^{-1}(\{y\})$  is regular w.r.t.\ $q.$  Let  $\gamma_j$ be the unique 
shortest geodesic with $\gamma_j(0)=q$ and $\gamma_j(1)=x_j.$
Since the first conjugate point along a geodesic never occurs  before the cut point, we have
$\lambda(\dot{\gamma_j}(0))<\pi^2$ for all $\lambda\in\Spec$ and all $j\in\{1,\dots, 2^r\}.$
Recall that $\Sing\cup\theta(\Sing)$
 is closed in $A$. 
 In view of Equation (\ref{EQ: Jacobi fields 1}) we want that $y=\Exp_q(Y)\in 
 A\setminus (\Sing\cup\theta(\Sing))$ satisfies
  \begin{equation}\label{EQ: Regularity 2}
   \lambda(\dot{\gamma_j}(0))\in \left]0,\pi^2\right[\setminus\left\{\nicefrac{\pi^2}{4}\right\} 
   \quad \text{for all}\; \lambda\in\Spec \; \text{and any}\; j\in\{1,\dots,2^r\}.
 \end{equation}
 This can be achieved by replacing $Y$ with $\tau Y$ for some $\tau\in\R$ close to $1$ if necessary.\par
 The decomposition
 of $T_qP$  in (\ref{EQ: Decomposition of TP}) allows us to choose
a positively oriented basis 
 $V_1,\dots, V_d$ of $T_qP,$ $d=\dim(P),$ such that 
 \begin{itemize}
  \item  $V_1,\dots, V_r\in T_qA;$ 
  \item $V_j\in E_{\lambda}$ for some $\lambda\in\Spec$ if $j\in\{r+1,\dots, d\}.$
 \end{itemize}
The cardinality of $\{V_1,\dots, V_d\}\cap E_{\lambda}$ is obviously the dimension
of $E_{\lambda}.$
Let us fix $j\in\{1,\dots 2^r\},$ and let
$V^{\|}_k$ be the parallel vector field along 
$\gamma_j$  with $V^{\|}_k(0)=V_k.$ 
Since parallel translation preserves the orientation, 
$V^{\|}_1,\dots, V^{\|}_d$ is a positively oriented frame field along 
$\gamma_j.$
Let $J_k$ be the Jacobi field along 
$\gamma_j$ with $J_k(0)=0$ and $\big(\nabla_{\dot{\gamma_j}} J_k\big)(0)=V_k.$ 
As $\lambda(\dot{\gamma_j}(0))<\pi^2$ for each $\lambda\in\Spec,$  Equation (\ref{EQ: Jacobi fields 1})
 shows that $J_1(1),\dots, J_d(1)$ is a positively oriented basis 
of $T_{x_j}P.$  Equations (\ref{EQ: differential theta 1}) and (\ref{EQ: Jacobi fields 1})
yield
\begin{equation}
 \label{EQ: Differential theta}
 d_{x_j}\theta(J_k(1))=J_k(2)
 =\left\{\begin{array}{ccl}
               \frac{1}{\sqrt{\lambda(\dot{\gamma_j}(0))}}\sin\big(2\sqrt{\lambda(\dot{\gamma_j}(0))}\big)\cdot V^{\|}_k(2), & 
               \text{if} &
               k>r,\; 
               V_k\in E_{\lambda}\\
               2\cdot V^{\|}_k(2) ,&\text{if} &
              k\in\{1,\dots,r\}\\
              \end{array}\right. 
\end{equation}
for $k\in\{1,\dots, d\}.$ 
From   (\ref{EQ: Regularity 2}) and  (\ref{EQ: Differential theta}) we now deduce:

\begin{thm}
\label{THM: Mapping degree}
The point $y$ chosen above is a  regular value of $\theta,$ and 
 the mapping 
 degree of $\theta$ is 
$$\deg(\theta)=\sum\limits_{j=1}^{2^r}(-1)^{\varepsilon_j},$$
with 
$$
 \varepsilon_j\; :\equiv\; \sum\limits_{\substack{\lambda\in\Spec\\ 
 \lambda(\dot{\gamma_j}(0))>\nicefrac{\pi^2}{4}}}
\dim(E_{\lambda})
\; \equiv\; 
\sum\limits_{\substack{\lambda\in\Spec\\ \lambda(\dot{\gamma_j}(0))>\nicefrac{\pi^2}{4}\\
\dim(E_{\lambda})\, \mathrm{odd}}}
\dim(E_{\lambda})\quad \pmod 2. 
$$
\end{thm}

\begin{rem}
The elements of $\Spec$ are actually the squares of the \emph{roots} of 
the symmetric space $P$ and the dimensions of the spaces $E_{\lambda}$
are the \emph{multiplicities} of these roots. So
Theorem \ref{THM: Mapping degree} provides an expression of 
$\deg(\theta)$ in terms of the root system with multiplicities of 
$P.$
\end{rem}


\subsection{Examples}

Let  $P=\LieG/\LieK$ be of \emph{splitting rank}, i.\ e.\
$\Rank(\LieG)=\Rank(\LieK)+\Rank(P),$
or, 
equivalently, $\dim(E_{\lambda})$ is even for any 
$\lambda\in\Spec$ (see e.g.\ 
\cite[Thm.\ 6.1, p.\ 429 ]{He} or \cite[Thm.\ 4.3, p.\ 81]{L-II}).
Then
$\varepsilon_j$ defined in Theorem \ref{THM: Mapping degree}
vanishes  for any $j\in \{1,\dots 2^r\}.$ Therefore
$\deg(\theta)=2^r.$ 

\begin{thm}
\label{THM: Rank split}
The canonical product of an oriented compact symmetric space of splitting rank
is a $\Gamma$-structure.
\end{thm}

\begin{rem}
 \begin{enumerate}[(i)]
  \item If $P$ is a compact Lie group with a bi-invariant metric, then $P$ is orientable and 
  all roots of the symmetric space $P$ have multiplicity equal to two 
  (see e.g.\  \cite[Thm. 4.4., p.\ 82]{L-II}). Thus  the canonical product on 
  $P$ (as a symmetric space)
  is a $\Gamma$-structure  different from its Lie theoretic product. 
  \item By \textsc{\'Elie Cartan's} classification of symmetric spaces (see e.g.\ 
  \cite[Chap.\ X]{He}) compact symmetric spaces of splitting rank
 are locally isomorphic to finite products of
  \begin{itemize}
   \item compact Lie groups (including flat tori),
   \item odd dimensional round spheres,
   \item $\LieSU_{2n}/\LieSp_n$ with $n\geq 3$ 
  \item $\LieE_6/\LieF_4.$
  \end{itemize}
  \item Theorem \ref{THM: Rank split} explains the form of the rational cohomology 
  of $\LieSU_{2n}/\LieSp_n$ given in 
   \cite[Thm.\ 6.7(1), p.\ 149]{MT}
   and of $\LieE_6/\LieF_4$ determined by \textsc{Araki} in \cite[Prop.\ 2.5]{Ar}.
 \end{enumerate}
\end{rem}


Next we recover the following result of \textsc{Albers}, \textsc{Frauenfelder} and 
\textsc{Solomon}: 

\begin{thm}[\cite{AFS}]
\label{THM: Lagrangian Grassmannian}
Let $P:=\LieU_n/\LieO_n$ with $n=2m+1$ and $m\in\N.$ 
Then
$\deg(\theta)
=2^{m+1},$ and the canonical product on $P$ is a $\Gamma$-structure.
\end{thm}

\begin{rem}
The space 
$P=\LieU_n/\LieO_n$ can be canonically identified with the 
space of all unoriented real
Lagrangian subspaces of the canonical symplectic space $\C^n.$ 
Let $I$ be the identity on $\C^n.$ 
Since $-I$  acts trivially on $P,$ $P$ can be written as $(\LieU_n/\{\pm I\})/(\LieO_n/\{\pm I\}).$
For $n$ odd $\LieO_n/\{\pm I\}\cong\LieSO_n$ is connected,
and $P$ satisfies the assumptions of 
Theorem \ref{THM: classification}.\par
\end{rem}

\begin{proof}
Since $\LieU_1/\LieO_1$ is isomorphic to $S^1,$ we may
assume that $m\geq 1.$ 
Since the Dynkin diagram of $\LieSU_n/\LieSO_n$ has type 
$\lieA_{n-1}$ (see e.g.\ \cite[Chap.\ X]{He}), the tangent space $T_qA$ of a 
flat $A\subset \LieU_n/\LieO_n$
can be identified with 
$\R^n$ 
such that 
$$\Spec\cong\big\{(e_j^*-e_k^*)^2:\; 1\leq j<k\leq n\big\},$$
where $e^*_1,\dots, e^*_n$ denote the duals
of the standard basis  
$e_1,\dots, e_n$ of $\R^n$ (see e.g.\ \cite[p.\ 462]{He})
and  
$$\LambdaL\cong\mathrm{span}_{\Z}(\pi e_1,\, \pi e_2,\dots, \pi e_n).$$\par

For 
$\frac{\pi}{8}\gg\xi_1>\xi_2>\dots >\xi_n>0$ 
we set
$Y:=-2(\xi_1, \dots, \xi_n)\in\R^n\cong T_qA$
 and $y:=\Exp_q(Y).$
 Recall that 
 $\theta^{-1}\big(\{y\}\big)=\big\{\Exp_q(X_{\delta}):\; \delta\in\{0,1\}^n\big\}$ with 
 $X_{\delta}:=\frac{1}{2}Y+\frac{\pi}{2} \delta.$
For $1\leq j<k\leq n$ we get 
\begin{equation}
 \label{EQ: Combin Lagr Grass}
(e_j^*-e_k^*)^2(X_{\delta})
=\big(\underbrace{\xi_k-\xi_j}_{\in\left]-\nicefrac{\pi}{2}, 0\right[}
+(\underbrace{\delta_j-\delta_k}_{\in\{-1,0,1\}})\cdot\nicefrac{\pi}{2}\big)^2.
\end{equation}
This shows that  $y$ is a regular value of $\theta.$ Moreover, we deduce from Equation (\ref{EQ: Combin Lagr Grass})
that $(e_j^*-e_k^*)^2(X_{\delta})>\nicefrac{\pi^2}{4}$ if and only if 
$\delta_j=0$ and $\delta_k=1.$ 
Following \cite[p.\ 934]{AFS}
we define $\varepsilon_{\delta}$ for $\delta\in\{0,1\}^n$ 
to be the number modulo 2 of all pairs $(j,k)$ with $1\leq j<k\leq n$ 
such that $\delta_j=0$ and $\delta_k=1.$
Since all roots of $P=\LieU_n/\LieO_n$ have multiplicity 
1 (see e.g.\ \cite[p.\ 532]{He}),
i.\ e.\  $\dim(E_{\lambda})=1$ for all 
 $\lambda\in\Spec,$ we get
 $$
\varepsilon_{\delta}\equiv\sum\limits_{\substack{
 \lambda\in\Spec\\ \lambda(X_{\delta})>\nicefrac{\pi^2}{4}}}
\dim(E_{\lambda})\quad \pmod 2$$
Thus
$\deg(\theta)=\sum\limits_{\delta\in\{0,1\}^n}(-1)^{\varepsilon_{\delta}}$
by Theorem \ref{THM: Mapping degree}.
These equations
also appear in \cite[p.\ 934]{AFS}. The combinatorial problem 
to calculate  explicitly $\deg(\theta)$  from these equations
is solved in 
\cite[Lem.\ 2.2]{AFS}, where one finds that
$\deg(\theta)
=2^{m+1}$ for
$n=2m+1,\; m\in\N.$
\end{proof}

\subsection{Proof of Theorem \ref{THM: classification}}

Just like compact Lie groups (see e.\ g.\ \cite[Thm.\ 4.29, p.\ 198]{Knapp}) compact symmetric spaces 
admit finite covers that split off flat factors:


\begin{lem}
\label{LEM: split covering}
 Every compact symmetric space $P$ is finitely covered by a product 
 $T\times \tilde{Q}$ of a flat torus $T$ and a simply connected compact symmetric space
 $\tilde{Q}.$ 
\end{lem}

\begin{proof}
The deck transformation group $\Delta$ of the universal cover $\tilde{P}$ of $P$  is a finitely
generated
discrete subgroup of the abelian centralizer $C_{\Iso(\tilde{P})}(\Trans(\tilde{P}))$
of the transvection group $\Trans(\tilde{P})$ of $\tilde{P}$ in its 
isometry group (see \cite[Lem.\ 1.2, p.\ 194]{Sakai} 
and \cite[Thm.\ 8.3.11, p.\ 244]{W}).
Since $\tilde{P}\cong\R^k\times \tilde{Q},$ where $\tilde{Q}$ is a simply connected compact 
symmetric space, 
the isometry group of $\tilde{P}$ 
splits as $\Iso(\tilde{P})=\Iso(\R^k)\times \Iso(\tilde{Q}).$\par
Thus any element of $\Delta$ has the form  $f\times g$ for 
some $f\in C_{\Iso(\R^k)}(\Trans(\R^k))\cong\R^k$ and
some $g\in C_{\Iso(\tilde{Q})}(\Trans(\tilde{Q})).$ 
Since $\tilde{Q}$ is a symmetric space of compact type 
$C_{\Iso(\tilde{Q})}(\Trans(\tilde{Q}))$ is finite. Let $N$ be 
a common multiple of the orders of elements of $C_{\Iso(\tilde{Q})}(\Trans(\tilde{Q})),$
then $g^N=e$ for all $g\in C_{\Iso(\tilde{Q})}(\Trans(\tilde{Q})).$
Since $\Delta$ is abelian, the $N$-th power is an endomorphism of $\Delta$ whose image $\Delta^N$
acts  trivially on the $\tilde{Q}$ factor of $\tilde{P}.$ 
As $\Delta^N$ has finite index in $\Delta,$ the space 
$\tilde{P}/\Delta^N\cong T\times\tilde{Q},$ which is the desired cover of $P,$ is compact. 
\end{proof}



\begin{lem}
\label{LEM: Riemannian products}
Let $P_1$ and $P_2$ be two compact oriented symmetric spaces.
 Then the canonical product on the Riemannian product $P_1\times P_2$ is a $\Gamma$-structure 
 if and only if the canonical products on $P_1$ and on $P_2$ are both
$\Gamma$-structures. 
\end{lem}

\begin{proof}
The Riemannian product $P_1\times P_2$ is again a compact  oriented symmetric space and the geodesic 
symmetries of $P_1\times P_2$ are
products of geodesic symmetries of $P_1$ and of 
$P_2.$
Therefore the canonical product $\Theta_{1\times 2}$ on $P_1\times P_2$ 
is the product of the canonical products $\Theta_{1}$ and $\Theta_{2}$ on 
$P_1$ and $P_2,$
$$\begin{array}{rcl}
 \Theta_{1\times 2}: (P_1\times P_2)\times (P_1\times P_2)&\to& P_1\times P_2,\\
((x_1,x_2),(y_1,y_2))&\mapsto &s_{(x_1,x_2)}(y_1,y_2)
=(\Theta_{1}(x_1,y_1), \Theta_{2}(x_2,y_2)).
\end{array}$$
With a  fixed origin $q=(q_1,q_2)\in P_1\times P_2$ we get
$$\theta_{1\times 2}(x_1,x_2):=\Theta_{1\times 2}((x_1,x_2),(q_1,q_2))
=(\Theta_{1}(x_1,q_1), \Theta_{2}(x_2,q_2))=(\theta_1\times \theta_2)(x_1,x_2).$$
The claim follows, since the mapping degree is multiplicative, that is 
$$\deg(\theta_{1\times 2})=\deg(\theta_1\times \theta_2)
=\deg(\theta_1)\deg(\theta_2).$$
\end{proof}

\begin{lem}
\label{LEM: Coverings}
Let $p:\hat{P}\to P$ be an orientation preserving  Riemannian covering
between two compact oriented symmetric spaces.
Then the canonical product on $\hat{P}$ is a $\Gamma$-structure 
 if and only if the canonical product on $P$ is a $\Gamma$-structure.
\end{lem}

\begin{proof}
The canonical products $\hat{\Theta}$ on $\hat{P}$ and 
$\Theta$ on $P$ are related by
$$\Theta\circ(p\times p)=p\circ \hat{\Theta}.$$
Let $\hat{q}\in\hat{P}$ be a chosen origin and  $q:=p(\hat{q}).$
Then the maps 
$\hat{\theta}=\hat{\Theta}^{\hat{q}}:\hat{x}\mapsto \hat{s}_{\hat{x}}(\hat{q})$
and $\theta=\Theta^q: x\mapsto s_x(q)$ satisfy
$p\circ \hat{\theta}=\theta\circ p.$
Since $p$ is a covering, we get 
$$\deg(p)\deg(\hat{\theta})=\deg(\theta)\deg(p),$$
where $\deg(p)$ coincides with  the number of sheets of $p.$
Division by $\deg(p)$ yields
$\deg(\hat{\theta})=\deg(\theta).$
\end{proof}


The rational cohomology of a $d$-dimensional connected closed manifold $M$
endowed with the cup product is a graded algebra 
$$H^*(M;\Q)=\bigoplus\limits_{k=0}^d H^k(M;\Q).$$
The minimal number of homogeneous generators of  $H^*(M;\Q)$ in degree $k\in\{1,\dots, d\}$
is 
$$\dim\big(H^k(M;\Q)\big)-\dim\Big(H^k(M;\Q)\cap 
\Big\langle \bigoplus\limits_{j=0}^{k-1} H^j(M;\Q)\Big\rangle_{\text{alg}}
\Big),$$
where $\Big\langle \bigoplus\limits_{j=0}^{k-1} H^j(M;\Q)\Big\rangle_{\text{alg}}$ 
is the smallest 
sub-algebra of $H^*(M;\Q)$ containing
$\bigoplus\limits_{j=0}^{k-1} H^j(M;\Q).$\par

K\"unneth's formula tells us that $H^*(M_1\times M_2;\Q)\cong H^*(M_1;\Q)\otimes H^*(M_2;\Q)$
for  connected closed manifolds $M_1$ and $M_2.$
Let $\{\alpha_1,\dots,\alpha_m\}$ be a minimal system of homogeneous generators 
of the algebra $H^*(M_1;\Q)$  and let $\{\beta_1,\dots,\beta_s\}$ be 
a minimal system of homogeneous generators of 
$H^*(M_2;\Q).$ 
Then $\{\alpha_1\otimes 1,\dots, \alpha_m\otimes 1,
1\otimes \beta_1,\dots 1\otimes \beta_s\}$ 
is a minimal system of homogeneous generators of the algebra
$H^*(M_1\times M_2;\Q).$ 
Therefore the algebra $H^*(M_1\times M_2;\Q)$ is generated by homogeneous elements
of odd degrees, if and only if this holds for both $H^*(M_1;\Q)$ and $H^*(M_2;\Q).$\par 
 

 We are now able 
 to prove Theorem \ref{THM: classification}. 
Note that compact symmetric spaces 
that satisfy the hypotheses of Theorem \ref{THM: classification} are orientable.
Furthermore simply connected compact symmetric spaces and their Riemannian products  
 with flat tori meet the assumptions 
 of Theorem \ref{THM: classification}.
Let $P$ be a symmetric space as in
Theorem \ref{THM: classification}, and suppose that 
$H^*(P;\Q)$ is an exterior algebra generated by homogeneous elements in
odd degrees. By Lemma \ref{LEM: split covering} there is a Riemannian product 
$$\hat{P}:=T\times \tilde{P}_1\times \dots \times \tilde{P}_m$$
of a flat torus $T$  (points and circles are  considered zero and one dimensional 
tori) 
and irreducible simply connected compact symmetric 
spaces $\tilde{P}_1,\dots, \tilde{P}_m$ that finitely covers $P.$

\begin{lem}
\label{LEM: cohomologies iso} 
The covering $p:\hat{P}\to P$ induces an isomorphism between the 
 graded $\Q$-algebras 
$H^*(P;\Q)$ and $H^*(\hat{P}; \Q).$ 
\end{lem} 

\begin{proof} 
We prove this claim in two different ways. 
First, let $\hat{q}\in \hat{P}$ and $q:=p(\hat{q}).$ 
Let $\hat{G}$ and $G$ denote the identity components of the isometry groups 
of $\hat{P}$ and $P.$ Since the isotropy subgroups $\hat{K}\subset \hat{G}$ 
of $\hat{q}$ and $K\subset G$ of $q$ are
are both connected, the projection $p$ 
identifies their linear isotropy actions.
By \cite[Thm.\ 8.5.8]{W} the real cohomology algebra
of any compact symmetric space $S$ is 
isomorphic to the algebra of those elements in $\bigwedge^*T_xS,\; 
x\in S,$ which are invariant under the action of the isotropy subgroup 
of $x$ in the identity component of the isometry group of $S.$
Thus $p$ induces an isomorphism $H^*(\hat{P};\R)\cong H^*(P;\R)$ 
and also an isomorphism of the rational cohomologies.\par

Second, the fundamental group $\pi_1(\hat{P})  < \pi_1(P)$ is abelian and acts trivially on 
$\pi_n(\hat{P}) = \pi_n(P)$ for $n > 1$, because $P = G / H$ with connected $H$ and
by Lemma \ref{simple_space}. Hence $\hat{P}$ is a simple space. Furthermore, because $p$ 
is a finite covering, the induced map $\pi_{*}(\hat{P}) \otimes \Q \to \pi_*(P)\otimes \Q$ is an isomorphism.
 By the Whitehead--Serre theorem for simple spaces (or inspecting the induced map between Postnikov 
decompositions of $\hat{P}$ and $P$) the induced map in rational cohomology is an isomorphism as well. 
\end{proof} 

Lemma \ref{LEM: cohomologies iso} and K\"unneth's formula imply
$$H^*(P;\Q)\cong H^*(T;\Q)\otimes H^*(\tilde{P}_1;\Q)\otimes\dots 
\otimes H^*(\tilde{P}_m; \Q).$$
By our previous  discussion all the algebras
 $H^*(\tilde{P}_j;\Q),\; j\in\{1,\dots, m\},$ 
are generated by homogeneous elements in odd degrees. 
To prove Theorem \ref{THM: classification} it now suffices 
by Lemmas \ref{LEM: Riemannian products} and 
\ref{LEM: Coverings} to show that the canonical products on $T$ and on $\tilde{P}_1,\dots,\tilde{P}_m$
are all $\Gamma$-structures.
As flat tori are symmetric spaces of splitting rank,  we are left to verify Theorem \ref{THM: classification} only
for irreducible simply connected 
compact symmetric spaces $P=G/K,$  where $G$ is the identity component of the isometry group of $P.$\par

Since the  rational cohomology of an \emph{inner} symmetric space, this is a space all of whose 
geodesic symmetries belong to the identity component $G$ of the isometry group, 
 has only  contributions in even degrees
  (see e.g.\ \cite[proof of Thm.\ 8.6.7]{W} and \cite[Thm.\ VII, p.\ 467]{GHV}), 
  we may assume that $P=G/K$  is an \emph{outer} symmetric space.
  Using the classification of symmetric spaces one could at this point determine all 
  outer symmetric spaces that satisfy the assumptions of 
  Theorem \ref{THM: classification} by 
checking  case-by-case the rational cohomology of such spaces given in \cite{Take} (see also \cite{MT, Spiv, GHV}). But we
prefer a  more conceptional approach that we essentially learned from \textsc{Oliver Goertsches.}\par
 Since $(\lieG,\lieK)$ is a \emph{Cartan pair} 
 (see \cite[pp.\ 448 \& 465]{GHV}) with $\rho:=\mathrm{rank}(\lieG)-\mathrm{rank}(\lieK)>0,$ 
 $H^*(G/K;\Q)$ is isomorphic to a tensor product of a  $2^{\rho}$ dimensional exterior algebra
  and a quotient of a symmetric algebra (see \cite[Thm.\ IV, p.\ 463]{GHV} and \cite[Thm.\ 3]{Kotschick}). 
Therefore
 $G/K$ satisfies the hypotheses of Theorem \ref{THM: classification} if and only if 
 $\dim(H^*(G/K;\Q))
 =2^{\rho}.$ By  \cite{Goertsches} 
this happens precisely if the number of Weyl chambers of $\lieG$ 
that intersect a given Weyl chamber of $\lieK$ is equal to one. 
Non-trivial intersections of Weyl chambers of $\lieG$ with Weyl chambers 
of $\lieK$ are called \emph{compartments} in \cite{EMQ}. \par
It turns out (see \cite[Sect.\ 3]{Murak} and \cite[Prop.\ 3.20(iii) \& Thm.\ 3.25]{Bur-Raw} and 
\cite[pp.\ 1128 \& 1129]{EMQ})
that $P$ only meets the assumptions of Theorem \ref{THM: classification},
if the involution of $\lieG$ associated with $P$ 
is a canonical extension $\tau_{\sigma}$
of an order two automorphism $\sigma$ of the Dynkin diagram of $\lieG$ 
as described in \cite[Sect.\ 3]{Murak} and \cite[p.\ 33]{Bur-Raw}.
The outer symmetric spaces of this kind are of splitting rank
or $\LieSU_{2m+1}/\LieSO_{2m+1}$ with $m\in\N$   (see \cite[Table p.\ 305]{Murak}).
In the first case 
Theorem \ref{THM: classification} follows directly from Theorem \ref{THM: Rank split}. For the second case
we note that $\LieU_{2m+1}/\LieO_{2m+1}$  is finitely
covered by $S^1\times \big(\LieSU_{2m+1}/\LieSO_{2m+1}\big).$ Theorem \ref{THM: Lagrangian Grassmannian}
together with
Lemmas \ref{LEM: Riemannian products} and \ref{LEM: Coverings} yield
$\deg(\theta)=2^m$ for  the canonical product of $\LieSU_{2m+1}/\LieSO_{2m+1}.$ Hence the canonical product  on $\LieSU_{2m+1}/\LieSO_{2m+1}$
is a $\Gamma$-structure. 



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{thebibliography}{xxxxx}



\bibitem[Ad]{Ad}
\textsc{J.\ F.\ Adams}, 
On the non-existence of elements of Hopf invariant one,
Ann.\ of Math.\   \textbf{72} (1960), 20--104


\bibitem[Ar]{Ar}
\textsc{S.\ Araki},
Cohomology modulo 2 of the compact exceptional group $\LieE_6$ and $\LieE_7,$
J.\ Math.\ Osaka City Univ., Ser.\ A \textbf{12} (1961), 43--65


\bibitem[AFS]{AFS}
\textsc{P.\ Albers, U. Frauenfelder} and \textsc{J.\ P.\ Solomon}, 
A $\Gamma$-structure on Lagrangian Grassmannians,
Comment.\ Math.\ Helv.\  \textbf{89} (2014), 929--936


\bibitem[Bo]{Bo}
\textsc{A.\ Borel}, 
Sur la cohomologie des espaces fibr\'es principaux et des espaces 
homog\`enes de groupes de Lie compacts,
Ann.\ Math.\ (2)  \textbf{57} (1953), 115--207

\bibitem[BR]{Bur-Raw}
\textsc{F.\ E.\ Burstall} and \textsc{J.\ H.\ Rawnsley}, 
Twistor theory for Riemannian symmetric spaces,
Lecture Notes in Mathematics \textbf{1424}, Springer-Verlag, Berlin 1990

\bibitem[Ca]{Ca}
\textsc{H. Cartan}, expos\'es 2 to 11, 
S\'eminaire Henri Cartan: Alg\`ebres d’Eilenberg--MacLane et homotopie, 7e ann\'ee, 
\'Ecole Normale Sup\'erieure (1954--55), 

\bibitem[Di]{D}
\textsc{J.\ Dieudonn\'e}, 
A history of algebraic and differential topology 1900--1960,
Birkhäuser, Boston 1989

\bibitem[EMQ]{EMQ}
\textsc{J.-H.\ Eschenburg,\; A.-L.\ Mare} and \textsc{P.\ Quast},
Pluriharmonic maps into outer symmetric spaces and a subdivision of Weyl chambers,
Bull.\ Lond.\ Math.\ Soc.\ \textbf{42} (2010), 1121--1133

\bibitem[Go]{Goertsches}
\textsc{O.\ Goertsches}, 
The equivariant cohomology of isotropy actions on symmetric spaces,
Doc.\ Math., J.\ DMV  \textbf{17} (2012), 79--94

\bibitem[GHV]{GHV}
\textsc{W.\ Greub, S.\ Halperin} and \textsc{R.\ Vanstone}, 
Connections, curvature, and cohomology, 
Vol. III: Cohomology of principal bundles and homogeneous spaces,
Academic Press, New York 1976

\bibitem[Ha1]{Ha1}
\textsc{A.\ Hatcher}, 
Algebraic topology,
Cambridge University Press, Cambridge 2002

\bibitem[Ha2]{Ha2}
\textsc{A.\ Hatcher}, 
Spectral Sequences in Algebraic Topology, preprint
available at: \url{https://www.math.cornell.edu/~hatcher/SSAT/SSATpage.html}

\bibitem[He]{He}
\textsc{S.\ Helgason}, 
Differential geometry, Lie groups, and symmetric spaces,
Academic Press, New York 1978

\bibitem[Ho1]{Ho1}
\textsc{H.\ Hopf}, 
Sur la topologie des groupes clos de Lie et de leurs généralisations,
C.\ R.\ Acad.\ Sci.\, Paris \textbf{208} (1939), 1266--1267


\bibitem[Ho2]{Ho2}
\textsc{H.\ Hopf}, 
\"Uber die Topologie der Gruppen-Mannigfaltigkeiten und ihre Verallgemeinerungen,
Ann.\ of Math.\ (2)  \textbf{42} (1941), 22--52

\bibitem[Kn]{Knapp}
\textsc{A.\ W.\ Knapp}, 
Lie groups beyond an introduction, 
Progress in Mathematics 140,  Birkh\"auser, Boston 1996

\bibitem[KT]{Kotschick}
\textsc{D.\ Kotschick} and \textsc{S.\ Terzi\'c},
On formality of generalized symmetric spaces,
Math.\ Proc.\ Camb.\ Philos.\ Soc.\   \textbf{134} (2003), 491--505

\bibitem[Le]{Le} 
\textsc{R.\ Lewis}, 
Homology and cell structures of nilpotent spaces, 
Trans.\ Amer.\ Math.\ Soc.\ {\bf 290} (1985), 747--760. 
% % \bibitem[Ish]{Ishi}
% % \textsc{K.\ Ishitoya}, 
% % Cohomology of the symmetric space $EI$,
% % Proc.\ Japan Acad., Ser.\ A  \textbf{53} (1977), 56--60


\bibitem[Lo1]{L-I}
\textsc{O.\ Loos}, 
Symmetric spaces I: general theory,
W.\ A.\ Benjamin, New York 1969

\bibitem[Lo2]{L-II}
\textsc{O.\ Loos}, 
Symmetric spaces II: compact spaces and classification,
W.\ A.\ Benjamin, New York 1969

\bibitem[MP]{MP}
\textsc{J.\ P.\ May} and \textsc{K.\ Ponto}, 
More concise algebraic topology: Localizations, completion, and model categories,
Chicago Lectures in Mathematics, Chicago 2012

\bibitem[Mc]{McCl}
\textsc{J. McCleary}, 
A user's guide to spectral sequences, 
Second edition, Cambridge University Press, Cambridge 2001. 
\bibitem[MT]{MT}
\textsc{M.\ Mimura} and \textsc{H.\ Toda}, 
Topology of Lie groups, I and II,
American Mathematical Society, Providence 1991

\bibitem[Mu]{Murak}
\textsc{S.\ Murakami}, Sur la classification des alg\`ebres de Lie r\'eelles et
simples, Osaka J.\  Math.\ 
\textbf{2} (1965), 291--307


\bibitem[Sa]{Sakai}
\textsc{T.\ Sakai}, 
Riemannian geometry, American Mathematical Society, Providence 1996

\bibitem[Se]{Se}
\textsc{J.\ P.\ Serre},  
Homologie modulo $2$ des complexes d'Eilenberg-MacLane, 
Comm. Math. Helv. {\bf 27} (1953), 198--232. 


\bibitem[Sh]{Sh} 
\textsc{H.\ Shiga}, 
Rational homotopy type and self maps, 
J. Math. Soc. Japan {\bf 31} (1979), 427--434. 


\bibitem[Sp]{Spiv}
\textsc{M.\ Spivak}, 
A comprehensive introduction to differential geometry, Vol.\ 5, 3rd edition,
Publish or Perish, Houston 1999

\bibitem[Su]{Sul} 
\textsc{D.\ Sullivan}, 
Infinitesimal computations in topology, Publ.\ Math.\ IHES \textbf{47} (1977),
269--331


\bibitem[Ta]{Take}
\textsc{M.\ Takeuchi}, On Pontrjagin classes of compact symmetric spaces, 
J.\ Fac.\ Sci., Univ.\ Tokyo, Sect.\ I 
\textbf{9} (1962), 313--328 

\bibitem[Wo]{W}
\textsc{J.\ A.\ Wolf}, 
Spaces of constant curvature, 5th edition,
Publish or Perish, Wilmington 1984



\end{thebibliography}

\end{document}


1602.06753.extracted_bib
[Ad]{Ad \textsc{J.\ F.\ Adams, On the non-existence of elements of Hopf invariant one, Ann.\ of Math.\ 72 (1960), 20--104
1602.06753.extracted_bib
[Ar]{Ar \textsc{S.\ Araki, Cohomology modulo 2 of the compact exceptional group $\LieE_6$ and $\LieE_7,$ J.\ Math.\ Osaka City Univ., Ser.\ A 12 (1961), 43--65
1602.06753.extracted_bib
[AFS]{AFS \textsc{P.\ Albers, U. Frauenfelder and \textsc{J.\ P.\ Solomon, A $\Gamma$-structure on Lagrangian Grassmannians, Comment.\ Math.\ Helv.\ 89 (2014), 929--936
1602.06753.extracted_bib
[Bo]{Bo \textsc{A.\ Borel, Sur la cohomologie des espaces fibr\'es principaux et des espaces homog\`enes de groupes de Lie compacts, Ann.\ Math.\ (2) 57 (1953), 115--207
1602.06753.extracted_bib
[BR]{Bur-Raw \textsc{F.\ E.\ Burstall and \textsc{J.\ H.\ Rawnsley, Twistor theory for Riemannian symmetric spaces, Lecture Notes in Mathematics 1424, Springer-Verlag, Berlin 1990
1602.06753.extracted_bib
[Ca]{Ca \textsc{H. Cartan, expos\'es 2 to 11, S\'eminaire Henri Cartan: Alg\`ebres d’Eilenberg--MacLane et homotopie, 7e ann\'ee, \'Ecole Normale Sup\'erieure (1954--55),
1602.06753.extracted_bib
[Di]{D \textsc{J.\ Dieudonn\'e, A history of algebraic and differential topology 1900--1960, Birkhäuser, Boston 1989
1602.06753.extracted_bib
[EMQ]{EMQ \textsc{J.-H.\ Eschenburg,\; A.-L.\ Mare and \textsc{P.\ Quast, Pluriharmonic maps into outer symmetric spaces and a subdivision of Weyl chambers, Bull.\ Lond.\ Math.\ Soc.\ 42 (2010), 1121--1133
1602.06753.extracted_bib
[Go]{Goertsches \textsc{O.\ Goertsches, The equivariant cohomology of isotropy actions on symmetric spaces, Doc.\ Math., J.\ DMV 17 (2012), 79--94
1602.06753.extracted_bib
[GHV]{GHV \textsc{W.\ Greub, S.\ Halperin and \textsc{R.\ Vanstone, Connections, curvature, and cohomology, Vol. III: Cohomology of principal bundles and homogeneous spaces, Academic Press, New York 1976
1602.06753.extracted_bib
[Ha1]{Ha1 \textsc{A.\ Hatcher, Algebraic topology, Cambridge University Press, Cambridge 2002
1602.06753.extracted_bib
[Ha2]{Ha2 \textsc{A.\ Hatcher, Spectral Sequences in Algebraic Topology, preprint available at: \url{https://www.math.cornell.edu/~hatcher/SSAT/SSATpage.html
1602.06753.extracted_bib
[He]{He \textsc{S.\ Helgason, Differential geometry, Lie groups, and symmetric spaces, Academic Press, New York 1978
1602.06753.extracted_bib
[Ho1]{Ho1 \textsc{H.\ Hopf, Sur la topologie des groupes clos de Lie et de leurs généralisations, C.\ R.\ Acad.\ Sci.\, Paris 208 (1939), 1266--1267
1602.06753.extracted_bib
[Ho2]{Ho2 \textsc{H.\ Hopf, \"Uber die Topologie der Gruppen-Mannigfaltigkeiten und ihre Verallgemeinerungen, Ann.\ of Math.\ (2) 42 (1941), 22--52
1602.06753.extracted_bib
[Kn]{Knapp \textsc{A.\ W.\ Knapp, Lie groups beyond an introduction, Progress in Mathematics 140, Birkh\"auser, Boston 1996
1602.06753.extracted_bib
[KT]{Kotschick \textsc{D.\ Kotschick and \textsc{S.\ Terzi\'c, On formality of generalized symmetric spaces, Math.\ Proc.\ Camb.\ Philos.\ Soc.\ 134 (2003), 491--505
1602.06753.extracted_bib
[Le]{Le \textsc{R.\ Lewis, Homology and cell structures of nilpotent spaces, Trans.\ Amer.\ Math.\ Soc.\ {290 (1985), 747--760. % %
1602.06753.extracted_bib
[Ish]{Ishi % % \textsc{K.\ Ishitoya, % % Cohomology of the symmetric space $EI$, % % Proc.\ Japan Acad., Ser.\ A 53 (1977), 56--60
1602.06753.extracted_bib
[Lo1]{L-I \textsc{O.\ Loos, Symmetric spaces I: general theory, W.\ A.\ Benjamin, New York 1969
1602.06753.extracted_bib
[Lo2]{L-II \textsc{O.\ Loos, Symmetric spaces II: compact spaces and classification, W.\ A.\ Benjamin, New York 1969
1602.06753.extracted_bib
[MP]{MP \textsc{J.\ P.\ May and \textsc{K.\ Ponto, More concise algebraic topology: Localizations, completion, and model categories, Chicago Lectures in Mathematics, Chicago 2012
1602.06753.extracted_bib
[Mc]{McCl \textsc{J. McCleary, A user's guide to spectral sequences, Second edition, Cambridge University Press, Cambridge 2001.
1602.06753.extracted_bib
[MT]{MT \textsc{M.\ Mimura and \textsc{H.\ Toda, Topology of Lie groups, I and II, American Mathematical Society, Providence 1991
1602.06753.extracted_bib
[Mu]{Murak \textsc{S.\ Murakami, Sur la classification des alg\`ebres de Lie r\'eelles et simples, Osaka J.\ Math.\ 2 (1965), 291--307
1602.06753.extracted_bib
[Sa]{Sakai \textsc{T.\ Sakai, Riemannian geometry, American Mathematical Society, Providence 1996
1602.06753.extracted_bib
[Se]{Se \textsc{J.\ P.\ Serre, Homologie modulo $2$ des complexes d'Eilenberg-MacLane, Comm. Math. Helv. {27 (1953), 198--232.
1602.06753.extracted_bib
[Sh]{Sh \textsc{H.\ Shiga, Rational homotopy type and self maps, J. Math. Soc. Japan {31 (1979), 427--434.
1602.06753.extracted_bib
[Sp]{Spiv \textsc{M.\ Spivak, A comprehensive introduction to differential geometry, Vol.\ 5, 3rd edition, Publish or Perish, Houston 1999
1602.06753.extracted_bib
[Su]{Sul \textsc{D.\ Sullivan, Infinitesimal computations in topology, Publ.\ Math.\ IHES 47 (1977), 269--331
1602.06753.extracted_bib
[Ta]{Take \textsc{M.\ Takeuchi, On Pontrjagin classes of compact symmetric spaces, J.\ Fac.\ Sci., Univ.\ Tokyo, Sect.\ I 9 (1962), 313--328
1602.06753.extracted_bib
[Wo]{W \textsc{J.\ A.\ Wolf, Spaces of constant curvature, 5th edition, Publish or Perish, Wilmington 1984