Pontrjagin-Thom maps and the homology of the moduli stack of stable curves

TL;DR

This paper investigates the homology of the moduli stack of stable curves, using Pontrjagin-Thom maps to reveal new torsion classes and establish surjectivity results in a degree range proportional to the genus.

Contribution

It introduces a novel application of Pontrjagin-Thom maps to the homology of the moduli stack, uncovering new torsion classes and proving surjectivity in specific degree ranges.

Findings

Maps are surjective on homology in a range proportional to genus.

Existence of many new torsion classes in the homology.

Homology of the moduli stack is better understood via these maps.

Abstract

We study the singular homology (with field coefficients) of the moduli stack of stable n-pointed complex curves of genus g (the Deligne-Mumford compactification). Each of its irreducible boundary components determines via the Pontrjagin-Thom construction a map to a certain infinite loop space whose homology is well understood. We show that these maps are surjective on homology in a range of degrees proportional to the genus. This implies the existence of many new torsion classes in the homology of the moduli stack.