Homology representations of compactified configurations on graphs applied to M_2,n

2021/09/07 by Christin Bibby, Melody Chan, Bibby, Christin +5 · 1 citation
Mathematics · #05C10 (primary) #14H10 #14Q05 #14T20 #55P65 #55R80 #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2109.03302

openalex publication_date 2021/09/07 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

Abstract

We obtain new calculations of the top weight rational cohomology of the moduli spaces M_2,n, equivalently the rational homology of the tropical moduli spaces Δ_2,n, as a representation of S_n. These calculations are achieved fully for all n≤ 10, and partially -- for specific irreducible representations of S_n -- for n≤ 22. We also present conjectures, verified up to n=22, for the multiplicities of the irreducible representations std_n and std_n⊗ sgn_n. We achieve our calculations via a comparison with the homology of compactified configuration spaces of graphs. These homology groups are equipped with commuting actions of a symmetric group and the outer automorphism group of a free group. In this paper, we construct an efficient free resolution for these homology representations, from which we extract calculations on irreducible representations one at a time, simplifying the calculation of these homology representations.

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• Lie algebra homology with coefficients tensor products of the adjoint representation in relative polynomial degree 2

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