Floer theory and its topological applications

Abstract

We survey the different versions of Floer homology that can be associated to three-manifolds. We also discuss their applications, particularly to questions about surgery, homology cobordism, and four-manifolds with boundary. We then describe Floer stable homotopy types, the related Pin(2)-equivariant Seiberg-Witten Floer homology, and its application to the triangulation conjecture.Comment: Notes based on the 14th Takagi Lectures at the University of Tokyo; to appear in the Japanese Journal of Mathematic