Analysis Seminar: Generalized Elmendorf’s Theorem in Context (Sofía Martínez Alberga, Bryn Mawr College)

Abstract: In general, the objective of algebraic topology is to classify spaces using some algebraic invariants or up to some notion of equivalence. In the area of equivariant homotopy theory, the goal is the same but now spaces equipped with a group action are considered and algebraic invariants of choice are homotopy groups. It turns out there is an analogous version of Whitehead’s theorem in the equivariant setting which in some sense motivates studying weak homotopy equivalences over homotopy equivalences. This talk will review some of these homotopical notions and introduce Elmendorf’s theorem. Proved in the eighties, this theorem sheds some light on how one can better understand equivariant homotopical notions as functors from the orbit category of the group to the category of topological spaces. Also, in this talk we will address how this perspective is used more modernly to understand better equivariant notions of other categories and to expand nonequivariant notions to equivariant ones.