Floer theory and its topological applications

C. Manolescu

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 ${\operatorname{Pin}(2)}$-equivariant Seiberg--Witten Floer homology, and its application to the triangulation conjecture.