Representations on the cohomology of M_0,n

2022/03/11 by Jinwon Choi, Choi, Jinwon, Young‐Hoon Kiem +3
Mathematics · #Homotopy and Cohomology in Algebraic Topology #Algebraic Geometry and Number Theory #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2203.05883

Abstract

The moduli space M_0,n of n pointed stable curves of genus 0 admits an action of the symmetric group S_n by permuting the marked points. We provide a closed formula for the character of the S_n-action on the cohomology of M_0,n. This is achieved by studying wall crossings of the moduli spaces of quasimaps which provide us with a new inductive construction of M_0,n, equivariant with respect to the symmetric group action. Moreover we prove that H^2k(M_0,n) for k≤ 3 and H^2k(M_0,n)⊕ H^2k-2(M_0,n) for any k are permutation representations. Our method works for related moduli spaces as well and we provide a closed formula for the character of the S_n-representation on the cohomology of the Fulton-MacPherson compactification ℙ¹[n] of the configuration space of n points on ℙ¹ and more generally on the cohomology of the moduli space M_0,n(ℙ^m-1,1) of stable maps.

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