Invariant homology classes

Question

Let $G$ be a finite group acting freely on a manifold $X$. What is the geometrical meaning of invariant homology classes $H_i(X,\mathbb Z)^G$? The same question for coinvariants $H_i(X,\mathbb Z)_G$.

Answer 1

I don't think these group have some nice description (apart from the definition) but they act as some approximation to homology of the factor.

More precisely. There is a map $H_G^\bullet(X)\to H^\bullet(X)^G$ and if the action of $G$ is free $H_G^\bullet(X)\cong H^\bullet(X/G)$, so we get a map $H(X/G)\to H(X)^G$. In general, it's far from being an isomorphism (take e.g. $X=EG$) but there is a spectral sequence $$ H^p(G;H^q(X))\implies H^{p+q}_G(X) $$ which can be used to extract some information about this map ($H^0(G;M)=M^G$, so $H(X)^G$ is exactly the leftmost column of $E_2$-term of this spectral sequence).

For example, over a field of char $0$ this spectral sequence degenerates, so $$ H^\bullet(X)^G\cong H^\bullet(X/G)\mod\text{torsion}. $$ (Oh, I prefer cohomology to homology, but everything is also true if we change all cohomology to homology [and invariants to coinvariants].)