David Gepner (University of Sheffield)
D.Gepner@sheffield.ac.uk It is well-known that orbifolds are to manifolds what Deligne-Mumford stacks are to schemes. That is, an orbifold is a manifold in which the points are allowed to have finite automorphism groups, together with some local triviality conditions. From the point of view of equivariant stable homotopy theory, however, it is more natural to consider orbispaces, i.e. things that locally look like a compact Lie group acting on a space.
In this talk I will present a homotopy theory of orbispaces, due to myself and A. Henriques, and explain how this theory reflects the stack-theoretic as well as the topological notions of equivalence. Since the homotopy theory of G-spaces, for any compact Lie group G, maps to the homotopy theory of orbispaces, it is natural to ask when equivariant cohomology theories, sufficiently functorial in the group of equivariance, give rise to orbispace cohomology theories. Time permitting, we will consider the case of elliptic cohomology.