Subrepresentations in the homology of finite covers of graphs

Abstract

Let $p:Y\to X$ 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 $H_1^{\text{comm}}(Y;\mathbb{C})\le H_1(Y;\mathbb{C})$ -- 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 $H_1^{\text{comm}}(Y;\mathbb{C})\ne \ker (p_{\ast })$.

14 pages. Comments welcome! Second version: Minor corrections and implementation of referee's recommendations (a proof of Corollary 2.5 (former Corollary 2.4) and an explanation of Example 4.1 were added), accepted for publication by Glasgow Mathematical Journal

References in corpus (1)

• Finite covers of graphs, their primitive homology, and representation theory