The tropicalization of the moduli space of curves II: Topology and applications

Abstract

We study the topology of the tropical moduli space parametrizing stable tropical curves of genus g with n marked points in which the bounded edges have total length 1, and prove that it is highly connected. Using the identification of this space with the dual complex of the boundary in the moduli space of stable algebraic curves, we give a simple expression for the top weight cohomology of M_{1,n} as a representation of the symmetric group and describe an explicit dual basis in homology consisting of abelian cycles for the pure mapping class group.

arXiv commentv2: 48 pages. Corrected statement and proof of Theorem 1.3, giving a lower bound for the connectivity of the tropical moduli space. This preprint will not be published. The main results will be subsumed in a series of papers, starting with arXiv:1805.10186

References in corpus (7)

• Representation theory and homological stability
• The tropicalization of the moduli space of curves
• Stabilization for mapping class groups of 3-manifolds
• On the tropical Torelli map
• The structure of the tautological ring in genus one
• Topology of the tropical moduli spaces $M_{2,n}$
• Parametrized Abel-Jacobi maps and abelian cycles in the Torelli group

Cited by in corpus (9)

• Topology of moduli spaces of tropical curves with marked points
• Assembling homology classes in automorphism groups of free groups
• Topology of the tropical moduli spaces $M_{2,n}$
• On the bordification of outer space
• Towards a tropical Hodge bundle
• Hyperelliptic graphs and the period mapping on outer space
• Getzler-Kapranov complexes and moduli stacks of curves
• Moduli spaces of codimension-one subspaces in a linear variety and their tropicalization
• Tropical Normal Functions -- Higher Abel-Jacobi Invariants of Tropical cycles