Universal Coefficients and Mayer-Vietoris Sequence for Groupoid Homology
Luciano Melodia
TL;DR
The work develops a Moore chain-model for the homology of ample, étale groupoids by using the compactly supported Moore complex of the nerve, and establishes a discrete universal coefficient theorem relating $H_n(\mathcal{G};A)$ to $H_n(\mathcal{G})$ whenever $A$ is discrete. It provides a robust computational toolkit, including long exact sequences for subgroupoid and quotient scenarios and a Mayer--Vietoris principle for clopen saturated covers, all tailored to the algebraic structure of compactly supported chains. The text also clarifies when Moore homology aligns with classical singular homology of the classifying space and underscores the necessity of discreteness for coefficient groups in the universal coefficient framework. Collectively, the results yield concrete, composable methods for calculating groupoid homology in the ample setting and show Kakutani and Morita-type invariances at the chain level. The theory thus connects topological groupoids, combinatorial nerve methods, and operator-algebraic considerations to produce explicit, transferable homological invariants for orbit-type groupoid models.
Abstract
We study homology of ample groupoids via the compactly supported Moore complex of the nerve. Let $A$ be a topological abelian group. For $n\ge 0$ set $C_n(\mathcal G;A) := C_c(\mathcal G_n,A)$ and define $\partial_n^A=\sum_{i=0}^n(-1)^i(d_i)_*$. This defines $H_n(\mathcal G;A)$. The theory is functorial for continuous étale homomorphisms. It is compatible with standard reductions, including restriction to saturated clopen subsets. In the ample setting it is invariant under Kakutani equivalence. We reprove Matui type long exact sequences and identify the comparison maps at chain level. For discrete $A$ we prove a natural universal coefficient short exact sequence $$0\to H_n(\mathcal G)\otimes_{\mathbb Z}A\xrightarrow{\ ι_n^{\mathcal G}\ }H_n(\mathcal G;A)\xrightarrow{\ κ_n^{\mathcal G}\ }\operatorname{Tor}_1^{\mathbb Z}\bigl(H_{n-1}(\mathcal G),A\bigr)\to 0.$$ The key input is the chain level isomorphism $C_c(\mathcal G_n,\mathbb Z)\otimes_{\mathbb Z}A\cong C_c(\mathcal G_n,A)$, which reduces the groupoid statement to the classical algebraic UCT for the free complex $C_c(\mathcal G_\bullet,\mathbb Z)$. We also isolate the obstruction for non-discrete coefficients. For a locally compact totally disconnected Hausdorff space $X$ with a basis of compact open sets, the image of $Φ_X:C_c(X,\mathbb Z)\otimes_{\mathbb Z}A\to C_c(X,A)$ is exactly the compactly supported functions with finite image. Thus $Φ_X$ is surjective if and only if every $f\in C_c(X,A)$ has finite image, and for suitable $X$ one can produce compactly supported continuous maps $X\to A$ with infinite image. Finally, for a clopen saturated cover $\mathcal G_0=U_1\cup U_2$ we construct a short exact sequence of Moore complexes and derive a Mayer-Vietoris long exact sequence for $H_\bullet(\mathcal G;A)$ for explicit computations.