Orthogonality and minimality in the homology of locally finite graphs

Abstract

Given a finite set E, a subset D ⊆ E (viewed as a function E → F2) is orthogonal to a given subspace F of the F2-vector space of functions E → F2 as soon as D is orthogonal to every ⊆-minimal element of F. This fails in general when E is infinite. However, we prove the above statement for the six subspaces F of the edge space of any 3-connected locally finite graph that are relevant to its homology: the topological, algebraic, and finite cycle and cut spaces. This solves a problem of Diestel (2010, arXiv:0912.4213).