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>

Binary factors of shifts of finite type

“…The first are those of Proietti and Yamashita [21,22], who show that the Smale space homology here agrees with the groupoid homology as studied by Matui. Further, Matui has recently adapted the results which are used in the next section for the K-theory of the C * -algebras to the case of groupoid homology: see [19]. These require the unit space to be totally disconnected and so only apply to parts (1), ( 2) and (5) of Theorem 5.1.…”