Floer theoretic invariants for 3- and 4-manifolds

TL;DR

This paper reviews various Floer theoretic invariants for 3- and 4-manifolds, discussing their relationships, known results, and open questions in the development of these topological tools.

Contribution

It provides a comprehensive overview of the current state and expected connections among different Floer invariants for manifolds.

Findings

Relationships among Seiberg-Witten, Heegaard Floer, and embedded contact homology invariants

Known results about their definitions and properties

Open questions and conjectures in Floer theory development

Abstract

Seiberg-Witten (Floer) theory, Ozsvath-Szabo's Heegaard Floer theory, Hutchings's embedded contact homology, in different stages of development, define (or are expected to define) packages of invariants for 3- and 4-manifolds (including manifolds with boundary and manifolds with certain types of corners). We describe what are known about their relationship, what are expected, and raise some questions along the way.