[Paper Review] The tropicalization of the moduli space of curves II: Topology and applications

Melody Chan, Søren Galatius+1 more|arXiv (Cornell University)|Apr 11, 2016

Algebraic Geometry and Number Theory41 references21 citations

TL;DR

This paper studies the topology of the tropical moduli space Δ_{g,n} parametrizing stable tropical curves of genus g with n marked points and total bounded edge length 1, proving its high connectivity and computing its rational homology as a representation of the symmetric group. Using the identification of Δ_{g,n} with the dual complex of the boundary in the compactified moduli space of algebraic curves, the authors derive explicit formulas for the top weight cohomology of M_{1,n} and construct an explicit dual basis of abelian cycles in homology for the pure mapping class group.

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.

Motivation & Objective

• To understand the topological structure of the tropical moduli space Δ_{g,n} parametrizing stable tropical curves of genus g with n marked points and total bounded edge length 1.
• To establish connectivity bounds for Δ_{g,n} using the contractibility of the repeated marking subcomplex.
• To compute the top weight cohomology of the moduli space M_{1,n} as a representation of the symmetric group S_n.
• To construct an explicit dual basis in homology for the pure mapping class group using abelian cycles.

Proposed method

• The authors identify Δ_{g,n} with the dual complex of the boundary divisor in the compactified moduli space ČM_{g,n}, enabling topological invariants of Δ_{g,n} to be interpreted as algebraic invariants of M_{g,n}.
• They use the canonical identification of the reduced rational homology of Δ_{g,n} with the top graded piece of the weight filtration on the rational cohomology of M_{g,n}.
• The contractibility of the subcomplex parametrizing tropical curves with repeated markings is established via explicit chain contraction using étale pullbacks and transfer maps.
• The weight spectral sequence on the logarithmic de Rham complex for M_{g,n} is used to relate relations among abelian cycles in homology to coboundaries on Δ_{g,n}.
• Explicit constructions of abelian cycles are achieved by analyzing the action of the dihedral group D_n on the vertices and edges of an n-gon, with the sign representation used to define the S_n-representation structure.
• The proof of acyclicity of the chain complex associated to the boundary complex relies on constructing a chain contraction using normalized transfer maps from étale covers.

Experimental results

Research questions

• RQ1 What is the connectivity of the tropical moduli space Δ_{g,n} for genus g > 1 and n ≥ 1?
• RQ2 How can the top weight cohomology of M_{1,n} be described as a representation of the symmetric group S_n?
• RQ3 What is the homotopy type of Δ_{1,n} for n ≥ 3, and how does it relate to spheres of dimension n−1?
• RQ4 How are abelian cycles for the pure mapping class group in middle homological degree related to coboundaries on Δ_{g,n}?
• RQ5 What is the role of the repeated marking subcomplex in determining the topology of Δ_{g,n}?

Key findings

• For g > 1, the tropical moduli space Δ_{g,n} is (n−5g+4)-connected, establishing a non-trivial connectivity bound.
• For n ≥ 3, Δ_{1,n} is homotopy equivalent to a wedge sum of (n−1)!/2 spheres of dimension n−1, while Δ_{1,1} and Δ_{1,2} are contractible.
• The top weight cohomology of M_{1,n} is non-zero only in degree i = n for n ≥ 3, and is isomorphic to Q^{(n−1)!/2} as a Q-vector space.
• As an S_n-representation, the top weight cohomology Gr^{W}_{2n}H^n(M_{1,n};Q) is isomorphic to the induced representation Ind_{D_n,φ}^{S_n} Res^{S_n}_{D_n,ψ} sgn, where φ and ψ are the dihedral actions on vertices and edges respectively.
• The authors construct an explicit dual basis in homology for the pure mapping class group using abelian cycles, with relations among them described via coboundaries on Δ_{g,n} using the weight spectral sequence.
• The repeated marking subcomplex in Δ_{g,n} is contractible for all g > 0 and n > 1, which is instrumental in proving the connectivity and homotopy type results.