Colloquium
3:30 p.m., Monday (Sept 25)
Math 100
Laura Scull
University of Michigan
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.
