Conjugation Cells [equivariant cohomology]

Question

I'm studying conjugation spaces, I have found in many sources that a conjugation cell is a conjugation space (without a proof). The widest approach that I have found so far is this paper (example 3.5)

But for me is not clear the following assertions in that example:

• $\rho: H^{2k}_C(D,S)\rightarrow H^{2K}(D,S)$ is an isomorphism
• Why the injectivity of $r: H^*_C(D,S) \rightarrow H^*_C(D^\tau,S^\tau)$ is needed?
• The conjugation equation $r\sigma(a) = \kappa(a)u^k$ holds trivially.

I really aprreciate if someone could explain me a little bit these facts.

Answer 1

As for your first question, the isomorphism $$ \rho\colon H_C^{2k}(D,S)\rightarrow H^{2k}(D,S) $$ follows from the spectral sequence $$ E_2^{pq}=H^q(C,H^p(D,S))\Rightarrow H_C^{p+q}(D,S). $$ Since $H^p(D,S)$ is nonzero only if $p=2k$, the spectral sequence degenerates at page $2$, and one gets an isomorphism $$ H^0(G,H^{2k}(D,S))\rightarrow H_C^{2k}(D,S). $$ Since $2k$ is even, the left-hand side is just the cohomology group $H^{2k}(D,S)$. It follows that the composition of the isomorphism with $\rho$ is the identity on $H^{2k}(D,S)$. This shows that $\rho$ is an isomorphism.