On the homological dimension of a Borel construction

Question

Let $M$ b a closed connected smooth manifold with fundamental group $\Gamma$. Suppose $G$ is a simply-connected Lie group that acts smoothly on $M$. Then the Borel construction $$M//G = M \times_G EG$$ has fundamental group $\Gamma$ as well.

Can the 1-truncation (i.e., the map classifying the universal cover) $$M//G \to B\Gamma$$ be non-trivial in rational cohomology in degrees bigger than the dimension of $M$?

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At first I thought perhaps one could produce a counterexample along the following lines:

Take $\Gamma < G$ a discrete group ($G$ some big simply-connected Lie group), and let $M = G/\Gamma$. Then $M//G = B\Gamma$, and the map in question is the identity.

But then I realized that the rational cohomological dimension of any such $\Gamma$ is at most $\text{dim}(G) - \text{dim}(K)$, where $K < G$ is a maximal compact subgroup, so this cannot possibly give a counterexample...

Answer 1

I think $f: M /\!\!/ G \to B\Gamma$ cannot be nontrivial on $\mathbb{Q}$-(co)homology in degrees beyond the dimension of $M$, because I think one can find a factorisation $$f_* : H_*(M /\!\!/ G ; \mathbb{Q}) \longrightarrow H_*(M / G ; \mathbb{Q}) \overset{\theta}\longrightarrow H_*(B\Gamma ; \mathbb{Q})$$ and the middle term vanishes for $* > dim(M)$ (and in fact often lower, depending on the dimension of the principal orbit).

Theorem 1.1 of

W. Browder, W.-C. Hsiang, G-actions and the Fundamental Group, Invent. Math. 65 (1981/82), no. 3, 411–424.

gives such a factorisation after precomposing with $H_*(M ; \mathbb{Q}) \to H_*(M /\!\!/ G ; \mathbb{Q})$, but looking at their proof it seems to prove the strengthening I have claimed above.

Their strategy is to consider $M$ as a stratified space, stratified by orbit type, and take the induced stratifications of $M \times EG$, $M/G$, and $M /\!\!/ G = (M \times EG)/G$. They then consider the functor $k$ sending a stratified space to its stratified 1-type. This allows us to form the commutative diagram $$\require{AMScd} \begin{CD} (M \times EG)/G @>>> k((M \times EG)/G) @>>> B\Gamma\\ @VVV@VV{\bar{t}}V \\ M/G @>>> k(M/G) \end{CD}$$ where the map to $B\Gamma$ is the canonical comparison between the stratified 1-type of $(M \times EG)/G$ and its ordinary 1-type. Now Lemma 3.4 (a) of the paper shows that $\bar{t}$ induces an isomorphism on rational homology, which gives the required factorisation.