Universal Coefficients and Mayer-Vietoris Sequence for Groupoid Homology

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 0 set C_n( G;A) := C_c( G_n,A) and define _n^A=_i=0^n(-1)^i(d_i)_*. This defines H_n( 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 The key input is the chain level isomorphism C_c( G_n, Z)_ ZA C_c( G_n,A), which reduces the groupoid statement to the classical algebraic UCT for the free complex C_c( G_, 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, Z)_ ZA C_c(X,A) is exactly the compactly supported functions with finite image. Thus Φ_X is surjective if and only if every f C_c(X,A) has finite image, and for suitable X one can produce compactly supported continuous maps X A with infinite image. Finally, for a clopen saturated cover G_0=U_1 U_2 we construct a short exact sequence of Moore complexes and derive a Mayer-Vietoris long exact sequence for H_( G;A) for explicit computations.

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