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.