Floer homologies for Lagrangian intersections and instantons

TL;DR

This paper discusses the development of Floer homologies for Lagrangian intersections and instantons, providing new tools for studying symplectic topology and gauge theory.

Contribution

It introduces Floer homologies in the context of Lagrangian intersections and instantons, extending the framework of invariants in low-dimensional topology.

Findings

Establishment of Floer homology theories for Lagrangian intersections.

Connections between Floer homologies and classical invariants like the Casson invariant.

New methods for analyzing 3-manifolds and symplectic geometry.

Abstract

In 1985 lectures at MSRI, A. Casson introduced an interesting integer valued invariant for any oriented integral homology 3-sphere Y via beautiful constructions on representation spaces (see [1] for an exposition). The Casson invariant \lambda(Y) is roughly defined by measuring the oriented number of irreducible representations of the fundamental group \pi_1(Y) in SU(2). Such an invariant generalized the Rohlin invariant and gives surprising corollaries in low dimensional topology.