Collapse of spectral sequence computing Equivariant cohomology

Question

I have already posted this question on math.stackexchanges but I got no answer and I decided to post it here.

Let us consider the fibration

$$ M\hookrightarrow EG\times_{\varphi}M\twoheadrightarrow BG $$ where $M$ is a $G-$space, $\varphi$ is the action of $G$ on $M$ and $EG\times_{\varphi}M$ is the diagonal quotient. It is required that $M$ is compact and path connected, $G$ is discrete and the action fixes at least one point $pt$. I would like to compute the equivariant cohomology $H_{G}^{\bullet}(M,\mathbb{Z})$. To do that, I can get the following information. Let us consider the Leray-Serre spectral sequence associated to the given fibration $$ E_{2}^{p,q}=H^{p}(BG,H^{q}(M,\mathbb{Z})) $$ and let us consider the terms $E_{2}^{p,0}$. In this case, because $M$ is path connected $H^{0}(M,\mathbb{Z})\simeq\mathbb{Z}$ and hence $$ E_{2}^{p,0}=H^{p}(BG,\mathbb{Z})\simeq H_{G}^{p}(pt,\mathbb{Z}) $$ because the action fixes the point $pt$. Introducing the reduced cohomology group $$ \widetilde{H}_{G}^{p}(M,\mathbb{Z}):=\ker i^{p} $$ where $i^{p}:H_{G}^{p}(M,\mathbb{Z})\twoheadrightarrow H_{G}^{p}(pt,\mathbb{Z})$ is the map in cohomology induced by the inclusion $i:pt\hookrightarrow M$, we have that $$ H_{G}^{p}(M,\mathbb{Z})\simeq\widetilde{H}_{G}^{p}(M,\mathbb{Z})\oplus H_{G}^{p}(pt,\mathbb{Z})\simeq\widetilde{H}_{G}^{p}(M,\mathbb{Z})\oplus E_{2}^{p,0}. $$ Thus $E_{2}^{p,0}$ is a subgroup of $H_{G}^{p}(M,\mathbb{Z})$. From that, I would like to deduce that $$ E_{2}^{p,0}=E_{\infty}^{p,0}\qquad\forall p\geqslant0, $$ i.e., $\textrm{Im}\,d_{2}^{p-2,1}=0$ for every $p\geqslant0$. I can not see why I can affirm that. Do you have some suggestions? Thanks in advance.

Answer 1

To prove this, use naturality of the Leray-Serre spectral sequence: the fixed point has a $G$-equivariant map $\ast \to M$ that induces a map of spectral sequences $$ H^p(G;H^q(M;\Bbb Z)) \to H^p(G;H^q(\ast;\Bbb Z)). $$ (This converges to the map $H^*_G(M) \to H^*_G(\ast)$.)

This map is compatible with the differentials, the map is an isomorphism when $p=0$, and the target is zero when $p > 0$. If an element $y$ on the 0-line were in the image of a differential, $y = d_r(x)$, then applying this map of spectral sequences we'd find $y = d_r(0)$ and hence $y=0$.