Borel equivariant homology of a suspension

Question

Let $G$ be a discrete group. For a $G$-CW complex $X$, let $H^G_{\bullet}(X)$ denote the Borel equivariant homology of $X$. There are also relative versions of this.

Here's my question. Let $X$ be a $G$-CW complex. The suspension $\Sigma X$ is then a $G$-CW complex in a natural way, and has two $G$-invariant base points coming from the two suspension points. Let $p_0$ be one of the suspension points. How can we compute $H^G_{\bullet}(\Sigma X,p_0)$ from $H^G_{\bullet}(X)$?

More generally, is there a good reference on Borel equivariant (co)homology where these sorts of foundational questions are worked out? All the references on equivariant homotopy theory I've looked at seem focused on different things, but Borel equivariant homology is what comes up in the things I'm working on right now.

Answer 1

I assume that by Borel equivariant homology of $X$ you mean the ordinary homology of the "Borel construction" $X\times_G EG$.

There is a homotopy cofibration sequence $$ X\times_G EG \to *\times_G EG \to \Sigma X\wedge_G EG_+. $$ It induces a long exact sequence in homology, which you can interpret as a long exact sequence of Borel homology groups $$\cdots \to H_n^G(X)\to H_n^G(*)\to H_n^G(\Sigma X, p_0)\to H_{n-1}^G(X) \to \cdots $$ Of course, $H_\bullet^G(*)$ is the same as the homology of $BG$, or the group homology of $G$.

If $X$ is a pointed $G$ space (in other words, if $X$ has a point $p$ fixed by $G$), then there is an isomorphism $$H_\bullet^G(X)\cong H_\bullet^G(p) \oplus H_\bullet^G(X, p).$$ In this case the long exact sequence simplifies to an isomorphism $$H_\bullet^G(\Sigma X, p_0)\cong H_{\bullet-1}^G(X, p).$$ This is just an equivariant version of the suspension isomorphism for homology.

But if $X$ does not have a $G$-fixed point then I think the long exact sequence above is the best general result you can have.