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Moving homology classes in finite covers of graphs (1509.09253v1)
Published 30 Sep 2015 in math.GT and math.GR
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.