Floer homology

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Floer homology<!--, named after ?????? Floer, --> 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 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...
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