Toward a minimal model for $H_\ast(\overline{\mathcal{M}})$

TL;DR

This paper constructs a minimal model for the homology of Deligne-Mumford moduli spaces, revealing the role of higher homology operations and providing explicit examples using Getzler's elliptic relation.

Contribution

It introduces an explicit construction of higher homology operations governing the minimal model of the homology of compactified moduli spaces.

Findings

Explicit higher homology operations constructed

First family of operations demonstrated using elliptic relation

Minimal model governed by these operations

Abstract

The modular operad $H_\ast(\overline{\mathcal{M}}_{g,n})$ of the homology of Deligne-Mumford compactifications of moduli spaces of pointed Riemann surfaces has a minimal model governed by higher homology operations on the open moduli spaces $H_\ast(\mathcal{M}_{g,n})$. Using Getzler's elliptic relation, we give an explicit construction of the first family of such higher operations.