Universal Coefficients and Mayer-Vietoris for Moore Homology of Ample Groupoids

TL;DR

This paper proves a universal coefficient theorem and a Mayer-Vietoris sequence for Moore homology of ample groupoids, providing tools for computing homology and highlighting the importance of discrete coefficients.

Contribution

It establishes a universal coefficient theorem and a Mayer-Vietoris sequence for Moore homology of ample groupoids, advancing computational methods.

Findings

Universal coefficient theorem relates homology with different coefficients.

Mayer-Vietoris sequence for Moore homology of groupoids.

Chain-level proof based on short exact sequences of chain complexes.

Abstract

We establish two structural results for Moore homology of ample groupoids. First, for every ample groupoid $\mathcal{G}$ and every discrete abelian coefficient group $A$, we prove a universal coefficient theorem relating the homology groups $H_n(\mathcal{G};A)$ to the integral Moore homology of $\mathcal{G}$. More precisely, we obtain a natural short exact sequence $$ 0 \longrightarrow H_n(\mathcal{G};\mathbb{Z})\otimes_{\mathbb{Z}} A \xrightarrow{\kappa_n^{\mathcal{G}}} H_n(\mathcal{G};A) \xrightarrow{\iota_n^{\mathcal{G}}} \operatorname{Tor}_1^{\mathbb{Z}}\bigl(H_{n-1}(\mathcal{G};\mathbb{Z}),A\bigr) \longrightarrow 0. $$ Second, for a decomposition of the unit space into clopen saturated subsets, we prove a Mayer-Vietoris long exact sequence in Moore homology. The proof is carried out at the chain level and is based on a short exact sequence of Moore chain complexes associated to the corresponding restricted groupoids. These results provide effective tools for the computation of Moore homology. We also explain why the discreteness of the coefficient group is essential for the universal coefficient theorem.