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Symplectic Geometry Seminar

Equivariant Floer Homology without Transversality (Semon Rezchikov)

Semon Rezchikov (IAS and Princeton)

Location:  Hill 705
Date & time: Thursday, 11 April 2024 at 1:20PM - 2:15PM

Title: Equivariant Floer Homology without Transversality

Abstract: The Cohen-Jones Segal construction assigns, to a smooth flow category, a stable homotopy type, or spectrum; in the case that the smooth flow category comes from a Morse- or Floer- theoretic construction, the homology of this spectrum will agree with the corresponding Morse- or Floer- homology group. However, even when bubbling does not occur, one can only hope to extract a virtually smooth flow category from Floer theory when one can achieve transversality. This makes it very challenging to extend the Cohen-Jones-Segal construction to the equivariant setting, where there are obstructions to achieving transversality using any small perturbation. In this talk I will explain a generalization of the Cohen-Jones-Segal construction which takes as input a virtually smooth flow category, in which the moduli spaces of morphisms are equipped with Kuranishi charts, rather than being smooth manifolds with corners. This construction assigns a spectrum to such a category, and immediately generalizes to the equivariant setting, producing a genuine G spectrum in that case. This G spectrum is generally not free, but can be reasoned about geometrically. I will discuss technical challenges that must be overcome in this construction, as well as potential applications. 

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