Floer theory and its topological applications

TL;DR

This survey explores various Floer homologies for three-manifolds, their topological applications, and recent advances including Floer stable homotopy types and their role in resolving the triangulation conjecture.

Contribution

It provides a comprehensive overview of Floer homology variants and introduces recent developments like Floer stable homotopy types and Pin(2)-equivariant Seiberg-Witten Floer homology.

Findings

Floer homologies have diverse applications in 3- and 4-manifold topology.

Recent developments have advanced understanding of the triangulation conjecture.

Floer stable homotopy types connect Floer homology to homotopy theory.

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.