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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 manifold] s.

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 three-dimensional manifold 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] .

** Symplectic Floer homology **

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