Moving homology classes in finite covers of graphs

TL;DR

This paper demonstrates that in finite covers of graphs, homology classes can be moved infinitely by lifts of homotopy equivalences, using representation theory to establish the existence of certain topological objects.

Contribution

It introduces a novel application of representation theory to prove the existence of topological objects in finite covers of graphs, enabling infinite movement of homology classes.

Findings

Existence of lifts of homotopy equivalences moving homology classes infinitely.

Use of representation theory to connect algebraic and topological properties.

Infinite orbits of homology classes under lifted homotopy maps.

Abstract

Let $Y\to X$ be a finite normal cover of a wedge of $n\geq 3$ circles. We prove that for any $v\neq 0\in H_1(Y;\mathbb{Q})$ there exists a lift $\widetilde{F}$ to $Y$ of a homotopy equivalence $F:X\to X$ so that the set of iterates $\{\widetilde{F}^d(v): d\in \mathbb{Z}\}\subseteq H_1(Y;\mathbb{Q})$ is infinite. The main achievement of this paper is the use of representation theory to prove the existence of a purely topological object that seems to be inaccessible via topology.