Orbifolds, Orbispaces and Global Homotopy Theory

TL;DR

This paper develops a framework using orthogonal spectra to represent orbifold cohomology theories, ensuring they satisfy key properties and are independent of orbifold presentation, with applications to various cohomology theories.

Contribution

It constructs orthogonal spectra for orbifolds' stable global homotopy types, unifying and extending orbifold cohomology theories with new representability results.

Findings

Orbifold cohomology theories are represented by orthogonal spectra.

The approach ensures cohomology groups are presentation-independent.

Examples include Borel, Bredon, and orbifold K-theory.

Abstract

Given an orbifold, we construct an orthogonal spectrum representing its stable global homotopy type. Orthogonal spectra now represent orbifold cohomology theories which automatically satisfy certain properties as additivity and the existence of Mayer-Vietoris sequences. Moreover, the value at a global quotient orbifold $M/G$ can be identified with the $G$-equivariant cohomology of the manifold $M$. Examples of orbifold cohomology theories which are represented by an orthogonal spectrum include Borel and Bredon cohomology theories and orbifold $K$-theory. This also implies that these cohomology groups are independent of the presentation of an orbifold as a global quotient orbifold.