Harer stability and orbifold cohomology

Nicola Pagani

Vol. 267 (2014), No. 2, 465–477

DOI: 10.2140/pjm.2014.267.465

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 of minimum 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 ${\mathcal{ℳ}}_{g,n}$.

Keywords

Harer stability, orbifold cohomology, moduli of curves, Chen–Ruan cohomology, automorphisms, age, twisted sector, inertia stack

Mathematical Subject Classification 2010

Primary: 14H10, 32G15, 55N32

Secondary: 14N35, 14D23, 14H37, 55P50

Milestones

Received: 16 August 2012

Revised: 27 October 2013

Accepted: 16 December 2013

Published: 11 May 2014