Moving homology classes in finite covers of graphs

Abstract

Let $Y\to X$ be a finite normal cover of a wedge of $n\ge 3$ circles. We prove that for any $v\ne 0\in H_1(Y;\mathbb{Q})$ there exists a lift $\tilde{F}$ to $Y$ of a homotopy equivalence $F:X\to X$ so that the set of iterates $\{{\tilde{F}}^d(v):d\in \mathbb{Z}\}\subseteq H_1(Y;\mathbb{Q})$ 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.

8 pages; 1 figure