Subrepresentations in the homology of finite covers of graphs
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Date
2023-09
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Journal Article
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Abstract
Let p : Y → X be a finite, regular cover of finite graphs with associated deck group G, and consider the first homology H₁(Y; 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 H₁(Y; C) on the other hand. We do so by studying certain subrepresentations in the G-representation H₁(Y; C).
The homology class of a lift of primitive element in Π₁(X) spans an induced subrepresentation in H₁(Y; C), and we show that this property is never sufficient to characterize such homology classes if G is Abelian. We study H₁ᶜᵒᵐᵐ(Y; C) ≤ H₁(Y; C) — the subrepresentation spanned by homology classes of lifts of commutators of primitive elements in Π₁(X). Concretly, we prove that the span of such homology class is isomorphic to the quotient of two induced representations. Furthermore, we construct examples of finite covers with H₁ᶜᵒᵐᵐ (Y; C) ≠ ker(p*).
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Journal / series
Volume
65 (3)
Pages / Article No.
582 - 594
Publisher
Cambridge University Press
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Subject
primitive homology; covers of graphs
Organisational unit
03491 - Burger, Marc (emeritus) / Burger, Marc (emeritus)