Equivariant cohomology and free $(\mathbb{Z} / 2)^n$-complexes

Abstract

[en] The field of transformation groups studies continuous actions of groups on topological spaces, in particular on CW-complexes. One of the fundamental questions that arises in this context is to determine those finite groups that can act effectively on a given topological space. A large amount of results are known about this issue, but it is not completely answered yet. Even in the case of abelian groups actions or elementary groups actions the question is highly nontrivial.

This project is devoted to a remarkable result regarding the description of those finite abelian groups that act freely on a CW-complex. The result states that if $X$ is a finite $C W$ complex and $(\mathbb{Z} / p)^n$ acts freely on $X$, with $p$ prime, then the sum of the lengths of the homology groups of $X$ with coefficients in $\mathbb{Z} / p$ is bounded below by $n+1$. Our study has been restricted to the case $p=2$, that was proved by Carlsson in 1983, with a modern approach based on cohomological methods.