Topology of moduli spaces of tropical curves with marked points

Abstract

We study a space of genus $g$ stable, $n$-marked tropical curves with total edge length $1$. Its rational homology is identified both with top-weight cohomology of the complex moduli space $M_{g,n}$ and with the homology of a marked version of Kontsevich's graph complex, up to a shift in degrees. We prove a contractibility criterion that applies to various large subspaces. From this we derive a description of the homotopy type of the tropical moduli space for $g=1$, the top weight cohomology of $M_{1,n}$ as an $S_n$-representation, and additional calculations for small $(g,n)$. We also deduce a vanishing theorem for homology of marked graph complexes from vanishing of cohomology of $M_{g,n}$ in appropriate degrees, and comment on stability phenomena, or lack thereof.

arXiv comment44 pages, 9 figures. This is a sequel to arXiv:1805.10186 and the second in a series of papers that subsumes arXiv:1604.03176. v2: Improved exposition. Final version, to appear in Facets of Algebraic Geometry: A Volume in Honour of William Fulton's 80th Birthday (Cambridge University Press)

References in corpus (10)

• Representation theory and homological stability
• The tropicalization of the moduli space of curves
• Stabilization for mapping class groups of 3-manifolds
• On a theorem of Kontsevich
• Stable cohomology of the mapping class group with symplectic coefficients and of the universal Abel-Jacobi map
• The structure of the tautological ring in genus one
• The tropicalization of the moduli space of curves II: Topology and applications
• Topology of the tropical moduli spaces $M_{2,n}$
• Oriented hairy graphs and moduli spaces of curves
• Factorization for stacks and boundary complexes

Cited by in corpus (13)

• Tropical curves, graph complexes, and top weight cohomology of M_g
• Tropical moduli spaces as symmetric Delta-complexes
• Graph complexes and Feynman rules
• Oriented hairy graphs and moduli spaces of curves
• The $S_n$-equivariant top weight Euler characteristic of $M_{g,n}$
• Grothendieck-Teichmueller group, operads and graph complexes: a survey
• Weight two compactly supported cohomology of moduli spaces of curves
• Hairy graphs to ribbon graphs via a fixed source graph complex
• Rational cohomology of the moduli space of trigonal curves of genus 5
• Configuration spaces on a wedge of spheres and Hochschild-Pirashvili homology
• Getzler-Kapranov complexes and moduli stacks of curves
• Automorphisms of tropical Hassett spaces
• Toward a minimal model for $H_{\ast }(\overline{\mathcal{M}})$