Intro to equivariant homotopy theory

In Fall 2024, the babytop seminar was on Hill-Hopkins-Ravenel’s resolution of the Kervaire invariant one problem. In this talk, I gave a brief introduction to equivariant homotopy theory. In the non-equivariant setting, homotopy theory is concerned with topological spaces up to weak equivalence. Before we can do equivariant homotopy theory, we need an equivariant notion of weak equivalence. Through a selection of examples, we try to motivate the relevant definitions. We then discuss Elmendorf’s Theorem and how it gives us a very nice, concrete model for the \(\infty\)-category of G-spaces as presheaves on the orbit category. We conclude by saying a few words about equivariance in families. This talk was based on Chapter 1 of Blumberg’s Burnside Category, and there are notes I prepared.