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

2016/10/27 by Benson Farb, Sebastian Hensel, Farb, Benson +1
Computer Science · Mathematics · #Biology #FOS: Mathematics #Genetics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homology (biology) #Homotopy and Cohomology in Algebraic Topology #Mathematics #Political science #Pure mathematics #Representation (politics) #Representation Theory (math.RT) #Topological and Geometric Data Analysis #math.GT #math.RT

paper · pdf · doi:10.48550/arxiv.1610.08819

24 pages, 1 figure

arxiv created 2016/10/27 · openalex publication_date 2016/10/27 · arxiv updated 2016/10/28 · openalex created_date 2016/11/04 · openalex updated_date 2026/07/28

Abstract

Consider a finite, regular cover Y→ X of finite graphs, with associated deck group G. We relate the topology of the cover to the structure of H₁(Y;ℂ) as a G-representation. A central object in this study is the \em primitive homology group H₁^prim(Y;ℂ)⊆ H₁(Y;ℂ), which is the span of homology classes represented by components of lifts of primitive elements of π₁(X). This circle of ideas relates combinatorial group theory, surface topology, and representation theory.

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