Weighted hyperprojective spaces and homotopy invariance in orbifold cohomology

TL;DR

This paper investigates the invariance of Chen-Ruan cohomology under specific homotopies, introducing T-representation homotopy, and demonstrates that weighted hyperprojective spaces are not S^1-representation homotopic despite being homotopy equivalent.

Contribution

The paper introduces T-representation homotopy and shows that Chen-Ruan cohomology is invariant under it, while providing examples where weighted hyperprojective spaces are not S^1-representation homotopic.

Findings

Chen-Ruan cohomology is a homotopy invariant in certain cases.

Weighted hyperprojective spaces are not S^1-representation homotopic.

Chen-Ruan cohomology rings differ over the rationals for these spaces.

Abstract

We show that Chen-Ruan cohomology is a homotopy invariant in certain cases. We introduce the notion of a T-representation homotopy, which is a stringent form of homotopy under which Chen-Ruan cohomology is invariant. We show that while hyperkahler quotients of the cotangent bundle to a complex vector space by a circle S^1 (here termed weighted hyperprojective spaces) are homotopy equivalent to weighted projective spaces, they are not S^1-representation homotopic. Indeed, we show that their Chen-Ruan cohomology rings (over the rationals) are distinct.