Orbifolds, Orbispaces and Global Homotopy 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.