Subrepresentations in the homology of finite covers of graphs

TL;DR

This paper explores the relationship between the representation theory of deck groups and the topological properties of homology classes in finite graph covers, revealing new insights into subrepresentations and their characterizations.

Contribution

It broadens the correspondence between deck group representations and homology classes, especially analyzing subrepresentations generated by lifts of primitive elements and commutators.

Findings

Homology classes of lifts of primitive elements span induced subrepresentations.

For Abelian groups, such classes do not fully characterize the subrepresentations.

Examples show the subrepresentation spanned by commutators can differ from the kernel of the induced map.

Abstract

Let $p \colon Y \to X$ be a finite, regular cover of finite graphs with associated deck group $G$, and consider the first homology $H_1(Y;\mathbb{C})$ of the cover as a $G$-representation. The main contribution of this article is to broaden the correspondence and dictionary between the representation theory of the deck group $G$ on the one hand, and topological properties of homology classes in $H_1(Y;\mathbb{C})$ on the other hand. We do so by studying certain subrepresentations in the $G$-representation $H_1(Y;\mathbb{C})$. The homology class of a lift of a primitive element in $\pi_1(X)$ spans an induced subrepresentation in $H_1(Y;\mathbb{C})$, and we show that this property is never sufficient to characterize such homology classes if $G$ is Abelian. We study $H_1^{\textrm{comm}}(Y;\mathbb{C}) \leq H_1(Y;\mathbb{C})$ -- the subrepresentation spanned by homology classes of lifts of commutators of primitive elements in $\pi_1(X)$. Concretely, we prove that the span of such a homology class is isomorphic to the quotient of two induced representations. Furthermore, we construct examples of finite covers with $H_1^{\textrm{comm}}(Y;\mathbb{C}) \neq \ker(p_*)$.