Long exact sequences of homology groups of \'etale groupoids

TL;DR

This paper establishes long exact sequences relating the homology groups of étale groupoids under subgroupoid and quotient conditions, with applications to SFT and hyperplane groupoids, advancing understanding of their algebraic topology.

Contribution

It introduces long exact sequences of homology groups for étale groupoids in subgroupoid and quotient scenarios, providing new tools for their algebraic analysis.

Findings

Existence of long exact sequences for subgroupoid cases.

Existence of long exact sequences for quotient groupoid cases.

Examples from SFT and hyperplane groupoids illustrating the theory.

Abstract

When a pair of \'etale groupoids $\mathcal{G}$ and $\mathcal{G}'$ 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, $\mathcal{G}'$ is assumed to be an open regular subgroupoid of $\mathcal{G}$. In the factor groupoid situation, we assume that $\mathcal{G}'$ is a quotient of $\mathcal{G}$ and the factor map $\mathcal{G}\to\mathcal{G}'$ 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.