GT seminar: Why twisted algebraic K-theory of spaces should enhance family Floer invariants

Abstract: In this talk, I will discuss how twisted algebraic K-theory of spaces should lift the usual notion of family Floer homology. That is, I want to present a heuristic that should generalize the invariants we obtained from twisted generating functions in cotangent bundles (arising from joint work with Abouzaid, Courte and Guillermou). These invariants, which are related to algebraic K-theory of spaces, were seen to be much stronger than the usual family Floer homology given by the family of fibers in the cotangent bundle (at least as treated so far). I will rely heavily on very specific examples for motivation. This heuristic generalization, motivates the need for a twisted version of Waldhausen's theorem, which would relate a twisted assembly map to a space of twisted h-cobordisms. If time permits, I will discuss ongoing joint work with Oldervoll trying to prove such a twisted Waldhausen theorem – relying on a category closely related to the category of flow categories.

This is a talk in our Geometry and Topology Seminar Series.