Universal Coefficients and Mayer-Vietoris Sequence for Groupoid Homology

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.

💡 Research Summary

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The paper develops a homology theory for ample (i.e., locally compact, totally disconnected, Hausdorff) étale groupoids by means of the compactly supported Moore complex of the nerve. For a topological abelian group A and each n ≥ 0 the authors set
(C_n(\mathcal G;A):=C_c(\mathcal G_n,A))
and define the boundary map
(\partial_n^A=\sum_{i=0}^n(-1)^i(d_i)*),
where (d_i) are the face maps of the nerve and ((d_i)*) the induced push‑forward on compactly supported functions. The resulting homology groups (H_n(\mathcal G;A)) are functorial with respect to continuous étale homomorphisms, compatible with standard reductions (e.g., restriction to saturated clopen subsets), and invariant under Kakutani equivalence. This invariance recovers Matui’s long exact sequences in a unified framework.

A central achievement is a universal coefficient theorem (UCT) for discrete coefficient groups A. The key observation is the chain‑level isomorphism
(C_c(\mathcal G_n,\mathbb Z)\otimes_{\mathbb Z}A\cong C_c(\mathcal G_n,A)).
Since the complex (C_c(\mathcal G_\bullet,\mathbb Z)) consists of free ℤ‑modules, the classical algebraic UCT applies verbatim, yielding a 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}(H{n-1}(\mathcal G),A)\to0).
The maps (ι_n^{\mathcal G}) and (κ_n^{\mathcal G}) are described explicitly at the chain level, providing a concrete computational tool.

For non‑discrete coefficients the above isomorphism fails. The authors introduce the natural map
(\Phi_X:C_c(X,\mathbb Z)\otimes_{\mathbb Z}A\to C_c(X,A))
and identify its image as precisely the compactly supported functions with finite image. Consequently, (\Phi_X) is surjective if and only if every compactly supported continuous map (X\to A) has finite image—a condition that typically fails for locally compact totally disconnected spaces with a basis of compact open sets. This obstruction explains why a universal coefficient theorem does not hold in general for non‑discrete coefficients.

The paper also constructs a short exact sequence of Moore complexes for a clopen saturated cover (\mathcal G_0=U_1\cup U_2):
(0\to C_\bullet(\mathcal G|{U_1\cap U_2};A)\to C\bullet(\mathcal G|{U_1};A)\oplus C\bullet(\mathcal G|{U_2};A)\to C\bullet(\mathcal G;A)\to0).
Passing to homology yields a Mayer–Vietoris long exact sequence for (H_\bullet(\mathcal G;A)). The authors illustrate the utility of this sequence with explicit calculations for transformation groupoids, AF‑groupoids, and examples arising from Kakutani equivalence, thereby recovering and extending Matui’s exact sequences.

In summary, the work provides a robust homology theory for ample étale groupoids that aligns with classical algebraic topology tools when coefficients are discrete, clarifies the limitations for non‑discrete coefficients, and equips researchers with Mayer–Vietoris machinery for practical computations. These results open new avenues for studying invariants of groupoid C(^*)‑algebras, dynamical systems, and non‑commutative geometry.