Harer stability and orbifold cohomology

TL;DR

This paper explores the orbifold cohomology of moduli spaces of curves, revealing that as genus grows large, the orbifold cohomology simplifies to ordinary cohomology, with explicit formulas for twisted sectors.

Contribution

It provides a combinatorial formula for the age of twisted sectors in algf4 moduli spaces and shows the orbifold cohomology converges to ordinary cohomology as genus increases.

Findings

Orbifold cohomology reduces to ordinary cohomology as genus algf4 increases.

The minimal age twisted sector is the hyperelliptic sector with Weierstrass points.

Age of the hyperelliptic sector equals half its codimension.

Abstract

In this paper we review the combinatorics of the twisted sectors of $\mathcal{M}_{g,n}$, and we exhibit a formula for the age of each of them in terms of the combinatorial data. Then we show that orbifold cohomology of $\mathcal{M}_{g,n}$ when $g \to \infty$ reduces to its ordinary cohomology. We do this by showing that the twisted sector with minimal age is always the hyperelliptic twisted sector with all markings in the Weierstrass points; the age of the latter moduli space is just half its codimension in $\mathcal{M}_{g,n}$.