Seiberg-Witten Floer Theory and Twisted Parametrised Spectra

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 once and for all. Later on, Khandawit, Lin, and Sasahira defined Seiberg--Witten Floer (ind/pro)-spectra for other classes of 3-manifolds, under suitable restrictions. How to construct Seiberg--Witten Floer spectra for all 3-manifolds is still open.

In this talk, we sketch how Seiberg-Witten Floer spectra can be constructed as twisted parametrised spectra. These are generalisations of parametrised spectra that were introduced by Douglas in his PhD thesis. We give an introduction to twisted parametrised spectra and explain how Seiberg-Witten Floer theory naturally gives rise to such objects. This is work in progress joint with S. Behrens and T. Kragh.