Universal Coefficients and Mayer-Vietoris for Moore Homology of Ample Groupoids
This work addresses computational challenges in algebraic topology for researchers studying groupoids, offering incremental tools for homology calculations.
The paper tackles the problem of computing Moore homology for ample groupoids by establishing a universal coefficient theorem and a Mayer-Vietoris sequence, providing exact sequences that relate homology groups with different coefficients and enable decomposition-based calculations.
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 $$ 0 \longrightarrow H_n(\mathcal{G};\mathbb{Z})\otimes_{\mathbb{Z}} A \xrightarrow{κ_n^{\mathcal{G}}} H_n(\mathcal{G};A) \xrightarrow{ι_n^{\mathcal{G}}} \operatorname{Tor}_1^{\mathbb{Z}}\bigl(H_{n-1}(\mathcal{G};\mathbb{Z}),A\bigr) \longrightarrow 0. $$ 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.