Equivariant Homotopy Theory: Classic Constructions in a Modern Setting

Equivariant homotopy theory studies spaces with an action of a compact Lie group. This is a much richer geometric object than the space alone, and the challenge is to adapt the techniques of algebraic topology to account for and take advantage of the extra structure provided by the group action. To introduce the issues arising, I will discuss the problem of defining equivariant cohomology. I will begin with a sketch of the usual idea of cohomology, which involves assembling spaces by gluing together simple pieces. Then I will look at methods for adapting cohomology to describe the action of a group. I will present two approaches: first, the classical use of bundles, and second, the more recent application of category theory. We will see some advantages and limitations to each.

My research is on algebraic models for rational equivariant homotopy theory. Such models have been developed for the actions of finite groups using category theory; however, these do not generalize to other groups. I have developed an algebraic model for actions of the circle group S^1. I will show how the model uses both the classical and the modern approaches discussed, combining bundles with categorical language. Finally, I will show that equivariant models, and particularly circle actions, are of interest by mentioning some relevant current research from algebraic topology, differential geometry and algebraic geometry.