Universal Coefficients and Mayer-Vietoris Sequence for Groupoid Homology

arXiv:2602.08998v1 Announce Type: cross
Abstract: We study homology of ample groupoids via the compactly supported Moore complex of the nerve. Let $A$ be a topological abelian group. For $nge 0$ 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 ‘etale 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 $$0to H_n(mathcal G)otimes_{mathbb Z}Axrightarrow{ iota_n^{mathcal G} }H_n(mathcal G;A)xrightarrow{ kappa_n^{mathcal G} }operatorname{Tor}_1^{mathbb Z}bigl(H_{n-1}(mathcal G),Abigr)to 0.$$ The key input is the chain level isomorphism $C_c(mathcal G_n,mathbb Z)otimes_{mathbb Z}Acong C_c(mathcal G_n,A)$, 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 $Phi_X:C_c(X,mathbb Z)otimes_{mathbb Z}Ato C_c(X,A)$ is exactly the compactly supported functions with finite image. Thus $Phi_X$ is surjective if and only if every $fin C_c(X,A)$ has finite image, and for suitable $X$ one can produce compactly supported continuous maps $Xto A$ with infinite image. Finally, for a clopen saturated cover $mathcal G_0=U_1cup U_2$ 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.