The first uniformly finite homology group with coefficients in ℤ and a characterisation of its vanishing in the transitive case

2020/01/14 by Bottinelli, Rémi, Kaiser, Tom
#Algebraic Topology (math.AT) #Combinatorics (math.CO) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2001.04857

Abstract

We study the first uniformly finite homology group of Block and Weinberger for uniformly locally finite graphs, with coefficients in ℤ and ℤ₂. When the graph is a tree, or coefficients are in ℤ₂, a characterisation of the group is obtained. In the general case, we describe three phenomena that entail non-vanishing of the group; their disjunction is shown to also be necessary for non-vanishing in the case of transitive graphs.

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