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Let me give an answer for $R=\mathbf Z$, the ring of integers, and let us translate to sheaf cohomology. Your long exact sequence comes from the short exact sequence $$ 0 \rightarrow j_! \mathbf Z \ri …

This is more generally true for semialgebraic subsets of $\mathbf R^n$ and follows from the fact that they are conical at infinity (see Bochnak, Coste, Roy: Real algebraic geometry, Corollary 9.3.7, p …

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^ …

In order to understand the isomorphism, I would translate everything to cohomology of sheaves and then use both Grothendieck's spectral sequences that converge to the same equivariant cohomology group …

I believe that your quotient space can be seen as the quotient of the $3$-sphere $S^3$ in $\mathbf C^2$ by the action of complex conjugation. The $3$-sphere $S^3$ can be identified with the join $S^1\ …

Did you try to use the double $T$ of the surface $S$? Any fixed point-free action of $\mathbf Z/n$ on $S$ induces a fixed point-free action on the closed orientable surface $T$. Moreover, the induced …

Using a little bit of real algebraic geometry, there is a conceptual proof at least in the critical case $\chi=-1$, i.e. the case you're talking about explicitly. Indeed, let $S$ be a compact connecte …