Long exact sequences of homology groups of étale groupoids

Abstract

<p style='text-indent:20px;'>When a pair of étale groupoids <inline-formula><tex-math id="M1">\begin{document}$ \mathcal{G} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ \mathcal{G}' $\end{document}</tex-math></inline-formula> on totally disconnected spaces are related in some way, we discuss the difference of their homology groups. More specifically, we treat two basic situations. In the subgroupoid situation, <inline-formula><tex-math id="M3">\begin{document}$ \mathcal{G}' $\end{document}</tex-math></inline-formula> is assumed to be an open regular subgroupoid of <inline-formula><tex-math id="M4">\begin{document}$ \mathcal{G} $\end{document}</tex-math></inline-formula>. In the factor groupoid situation, we assume that <inline-formula><tex-math id="M5">\begin{document}$ \mathcal{G}' $\end{document}</tex-math></inline-formula> is a quotient of <inline-formula><tex-math id="M6">\begin{document}$ \mathcal{G} $\end{document}</tex-math></inline-formula> and the factor map <inline-formula><tex-math id="M7">\begin{document}$ \mathcal{G}\to\mathcal{G}' $\end{document}</tex-math></inline-formula> 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.</p>