Long exact sequences of homology groups of étale groupoids

Abstract

When a pair of étale groupoids $\mathcal{G}$ and ${\mathcal{G}}^{\prime }$ on totally disconnected spaces are related in some way, we discuss the difference of their homology groups. More specifically, we treat two basic situations. In the subgroupoid situation, ${\mathcal{G}}^{\prime }$ is assumed to be an open regular subgroupoid of $\mathcal{G}$. In the factor groupoid situation, we assume that ${\mathcal{G}}^{\prime }$ is a quotient of $\mathcal{G}$ and the factor map $\mathcal{G}\to {\mathcal{G}}^{\prime }$ is proper and regular. For each, we show that there exists a long exact sequence of homology groups. We present examples which arise from SFT groupoids and hyperplane groupoids.

36 pages