On the homology of partial group actions

Abstract

We study the partial group (co)homology of partial group actions using simplicial methods. We introduce the concept of universal globalization of a partial group action on a $K$-module and prove that, given a partial representation of $G$ on $M$, the partial group homology $H^{par}_{\bullet}(G, M)$ is naturally isomorphic to the usual group homology $H_{\bullet}(G, KG \otimes_{G_{par}} M)$, where $KG \otimes_{G_{par}} M$ is the universal globalization of the partial group action associated to $M$. We dualize this result into a cohomological spectral sequence converging to $H^{\bullet}_{par}(G,M)$.