[Paper Review] Graph cohomology and Kontsevich cycles

Kiyoshi Igusa|ArXiv.org|Mar 13, 2003

Homotopy and Cohomology in Algebraic Topology10 references4 citations

TL;DR

This paper establishes a precise correspondence between dual Kontsevich cycles in associative graph cohomology and polynomials in the adjusted Milnor-Morita-Mumford classes in the cohomology of the mapping class group. Using a rational chain homotopy equivalence between the cellular chain complex of ribbon graphs and the graph cohomology complex, it computes explicit formulas for the first two terms of the polynomial, verifying and extending conjectures by Arbarello and Cornalba and resolving a question posed by Kontsevich.

ABSTRACT

The dual Kontsevich cycles in the double dual of associative graph homology correspond to polynomials in the Miller-Morita-Mumford classes in the integral cohomology of mapping class groups. I explain how the coefficients of these polynomials can be computed using Stasheff polyhedra and results from my previous paper GT/0207042.

Motivation & Objective

• To clarify the relationship between Kontsevich cycles in associative graph homology and the Miller-Morita-Mumford classes in the cohomology of the mapping class group.
• To extend earlier results by Igusa and provide detailed proofs of the duality between graph cohomology and mapping class group cohomology.
• To compute the coefficients of the first two terms in the polynomial expression for the dual Kontsevich cycles in terms of adjusted Miller-Morita-Mumford classes.
• To verify and generalize conjectured formulas by Arbarello and Cornalba for the action of these cycles on the moduli space of curves.

Proposed method

• Constructs a rational chain homotopy equivalence between the cellular chain complex of the category of ribbon graphs and the integral graph cohomology complex.
• Uses the forest carrier construction to define an acyclic ${\mathbb{Z}}$-augmented chain complex over the graph cohomology complex.
• Applies the Conant-Vogtmann definition of graph orientation to determine the intrinsic orientation of Stasheff associahedra and justify sign conventions.
• Employs a cyclic set cocycle to evaluate Miller-Morita-Mumford classes on the Kontsevich cycles.
• Performs detailed sign and multiplicity calculations on specific ribbon graph configurations to derive coefficients.
• Derives a rational inverse map $\psi$ from graph cohomology to the cellular chain complex, establishing a rational homotopy equivalence.

Experimental results

Research questions

• RQ1 How do the dual Kontsevich cycles in associative graph cohomology relate to the Miller-Morita-Mumford classes in the cohomology of the mapping class group?
• RQ2 What is the explicit polynomial expression in the adjusted Miller-Morita-Mumford classes that represents the dual Kontsevich cycles?
• RQ3 How do the coefficients of this polynomial relate to the conjectures of Arbarello and Cornalba for the Poincaré duals of strata in the moduli space of curves?
• RQ4 What is the precise structure of the rational chain homotopy equivalence between the ribbon graph category and the graph cohomology complex?
• RQ5 Can the conjectured symmetry in the coefficients of the Kontsevich cycle formula be proven and extended to general $n,k$?

Key findings

• For $n \neq 1$, the dual Kontsevich cycle $[W_{n,1}^\ast]$ is given by $3(-2)^{n+3}(2n+1)!!(\widetilde{\kappa}_n\widetilde{\kappa}_1 - \widetilde{\kappa}_{n+1}) - (-2)^{n+2}(2n+5)!!\widetilde{\kappa}_{n+1}$.
• For $n = 1$, the formula is divided by 2, yielding $[W_{1,1}^\ast] = 72\widetilde{\kappa}_1^2 + 348\widetilde{\kappa}_2$, which matches known computations.
• The formula for $[W_{n,1}^\ast]$ agrees with the calculations of Arbarello and Cornalba, confirming their conjectured action of the cycles on boundary strata.
• The conjectured general formula for $[W_{n,k}^\ast]$ is verified for all $k \leq 7$, and the case $n=k=2$ yields $[W_{2,2}^\ast] = 7200\widetilde{\kappa}_2^2 + 159120\widetilde{\kappa}_4$, consistent with their results.
• The coefficient $[W_{1,1,1}^\ast] = 288\widetilde{\kappa}_1^3 + 4176\widetilde{\kappa}_1\widetilde{\kappa}_2 + 20736\widetilde{\kappa}_3$ is computed and shown to be consistent with the conjectured structure.