Floer homology

In mathematics, Floer homology is a tool for studying symplectic geometry and low-dimensional topology. Floer homology is a novel invariant that arises as an infinite-dimensional analogue of finite-dimensional Morse homology. Andreas Floer introduced the first version of Floer homology, now called symplectic Floer homology, in his proof of the Arnold conjecture in symplectic geometry. Floer also developed a closely related theory for Lagrangian submanifolds of a symplectic manifold. A third construction, also due to Floer, associates homology groups to closed three-dimensional manifolds using the Yang–Mills functional. These constructions and their descendants play a fundamental role in current investigations into the topology of symplectic and contact manifolds as well as (smooth) three- and four-dimensional manifolds.

Floer homology is typically defined by associating to the object of interest an infinite-dimensional manifold and a real valued function on it. In the symplectic version, this is the free loop space of a symplectic manifold with the symplectic action functional. For the (instanton) version for three-manifolds, it is the space of SU(2)-connections on a three-dimensional manifold with the Chern–Simons functional. Loosely speaking, Floer homology is the Morse homology of the function on the infinite-dimensional manifold. A Floer chain complex is formed from the abelian group spanned by the critical points of the function (or possibly certain collections of critical points). The differential of the chain complex is defined by counting the flow lines of the function's gradient vector field connecting fixed pairs of critical points (or collections thereof). Floer homology is the homology of this chain complex.

The gradient flow line equation, in a situation where Floer's ideas can be successfully applied, is typically a geometrically meaningful and analytically tractable equation. For symplectic Floer homology, the gradient flow equation for a path in the loopspace is (a perturbed version of) the Cauchy–Riemann equation for a map of a cylinder (the total space of the path of loops) to the symplectic manifold of interest; solutions are known as pseudoholomorphic curves. The Gromov compactness theorem is then used to show that the counts of flow lines defining the differential are finite, so that the differential is well-defined and squares to zero. Thus the Floer homology is defined. For instanton Floer homology, the gradient flow equation is exactly the Yang–Mills equation on the three-manifold crossed with the real line.