# GT Seminar with Alice Hedenlund (Uppsala university)

- Date: 1 December 2022, 13:15–14:15
- Location: Ångström Laboratory, room Å64119 and on zoom https://uu-se.zoom.us/j/68210068219
- Type: Seminar
- Lecturer: Alice Hedenlund (Uppsala university)
- Organiser: Matematiska institutionen
- Contact person: Noémie Legout

Welcome to this seminar held by Alice Hedenlund (Uppsala university) with title "Seiberg-Witten Floer Homotopy Types I: Twisted Spectra".

**Abstract:** Seiberg–Witten theory has played a central role in the study of smooth low-dimensional manifolds since their introduction in the 90s. Parallel to this, Cohen, Jones, and Segal asked the question of whether various types of Floer homology could be upgraded to the homotopy level by constructing (stable) homotopy types encoding Floer data. In 2003, Manolescu constructed Seiberg-Witten Floer spectra for rational homology 3-spheres, and in particular used these to settle the triangulation conjecture. A precursor to Manolescu's Seiberg-Witten Floer spectra, the Bauer-Furuta invariant, was moreover used by Furuta to make significant progress on the "11/8 Conjecture" which deals with what quadratic forms are realisable as the intersection form on a smooth 4-manifold.

This is the first talk of a two-part series reporting on joint work in progress with S. Behrens and T. Kragh. In this talk I will give an expository introduction to twisted spectra, which are essential for constructing Floer homotopy types in the situation where our infinite-dimensional manifold is "non-trivially polarised". Roughly, one could think of twisted spectra as arising as global sections of a bundle whose fibre is the (infinity-)category of spectra. I will also explain how these mathematical objects naturally appear in Seiberg-Witten Floer theory. Next week, Thomas will go into the more concrete constructions of twisted spectra from Seiberg-Witten Floer data using finite-dimensional approximation and Conley index theory.

See upcoming seminars in our seminar series on Geometry and Topology (GT).