Alice Hedenlund: Twisted spectra and Floer homotopy theory

Speaker
Alice Hedenlund, Norwegian University of Science and Technology

Abstract
In the 90s, Cohen- Jones, and Segal asked the question of whether various types of Floer homology theories could be upgraded to the homotopy level by constructing stable homotopy types encoding Floer data. They also sketched how one could construct these Floer homotopy types as (pro)spectra in the situation that the flow category involved is framed and when there is no bubbling. It has since been realized that the correct home for Floer homotopy types, in the non-frameable situation, is twisted spectra. This is a generalization of parametrized spectra that one can roughly think of as sections of bundles of categories whose fibre is the category of spectra. The aim of this talk is to give an introduction to twisted spectra and sketch how they show up naturally in Floer homotopy theory. This involves several separate, but related projects: joint work with T. Moulinos on the six-functor formalism of twisted spectra, joint work in progress with S. Behrens and T. Kragh on constructing twisted spectra from Seiberg-Witten Floer theory, and joint work in progress with T. P. Oldervoll on the relationship between structured flow categories and twisted spectra.