Moving homology classes in finite covers of graphs

Abstract

Let Y → X be a finite normal cover of a wedge of n ≥ 3 circles. We prove that for any nonzero v ∈ H ₁(Y; Q) there exists a lift $\widetilde F$ to Y of a basepoint-preserving homotopy equivalence F: X → X such that the set of iterates $\left\{ {{{\widetilde F}^d}\left( v \right)} \right\}:d \in \mathbb{Z} \subseteq {H_1}\left( {Y,\mathbb{Q}} \right)$ 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.