Floer Homology

Floer homology is a mathematical tool used in the study of symplectic geometry and low-dimensional topology. First introduced by Andreas Floer in his proof of the Arnold conjecture in symplectic geometry, Floer homology is a novel homology theory arising as an infinite dimensional analog of finite dimensional Morse homology. A similar construction, also introduced by Floer, provides a homology theory associated to three-dimensional manifolds. This theory, along with a number of its generalizations, plays a fundamental role in current investigations into the topology of three- and four-dimensional manifolds. Using techniques from gauge theory, these investigations have provided surprising new insights into the structure of three- and four-dimensional differentiable manifolds.

Floer homology is typically defined by associating an infinite dimensional manifold to the object of interest. In the symplectic version, this is the free loop space of a symplectic manifold, while in the original three-dimensional manifold (instanton) version, it is the space of SU(2)-connections on a three-dimensional manifold. Loosely speaking, Floer homology is the Morse homology computed from a natural function on this infinite dimensional manifold. This function is the symplectic action on the free loop space or the Chern–Simons function on the space of connections. A homology theory is formed from the vector space spanned by the critical points of this function. A linear endomorphism of this vector space is defined by counting the function's gradient flow lines connecting two critical points. Floer homology is then the quotient vector space formed by identifying the image of this endomorphism inside its kernel.

Instanton Floer homology is viewed as a generalization of the Casson invariant because the Euler characteristic of Floer homology is identified with Casson invariant.