Discrete Coefficients and Open Invariant Covers in the Homology of Ample Groupoids

arXiv:2603.20861v2 Announce Type: replace-cross Abstract: The homology of an ample groupoid is computed from the complex of compactly supported continuous functions on the nerve. Two hypotheses routinely imposed on this complex behave in opposite ways. We show that the comparison map from the integral chain complex tensored with the coefficient group to the chain complex with coefficients is always injective, and that it is surjective if and only if every compactly supported continuous function into the coefficient group is locally constant. The universal coefficient theorem for ample groupoids is therefore a discrete coefficient statement, and it already fails for the real numbers on the Cantor set. We then show that a cover of the unit space by two clopen invariant subsets forces the groupoid to split as a disjoint union of three clopen invariant reductions, so that the associated Mayer-Vietoris sequence has vanishing connecting maps and computes nothing. Dropping closedness repairs this, since two open invariant subsets covering the unit space give a Mayer-Vietoris sequence for every topological abelian coefficient group. For an integer action on a two point compactification of the integers we compute that sequence and find that its connecting map is an isomorphism of infinite cyclic groups, which no cover of a unit space by clopen invariant subsets can achieve.

Source: arXiv cs.LG | 2026-08-12