Orthogonality and minimality in the homology of locally finite graphs

TL;DR

This paper proves that orthogonality relations between certain subspaces of the edge space hold in 3-connected locally finite graphs, extending finite case results to infinite graphs and solving a known problem.

Contribution

It establishes orthogonality properties for homology-related subspaces in infinite graphs, specifically for 3-connected locally finite graphs, which was previously unresolved.

Findings

Orthogonality holds for six key subspaces in infinite graphs.

Extends finite graph homology results to infinite, locally finite graphs.

Solves a previously open problem in graph homology theory.

Abstract

Given a finite set $E$, a subset $D\sub E$ (viewed as a function $E\to \F_2$) is orthogonal to a given subspace $\FF$ of the $\F_2$-vector space of functions $E\to \F_2$ as soon as $D$ is orthogonal to every $\sub$-minimal element of $\FF$. This fails in general when $E$ is infinite. However, we prove the above statement for the six subspaces $\FF$ 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 [5]