Floer homology

In mathematics, Floer homology refers to a family of homology theories which share similar characteristics and are believed by experts to be closely related. Some of these theories are due directly to Andreas Floer, while others are derived or inspired by his work. They are all modelled upon Morse homology on finite dimensional manifolds, extending it to the case where the relevant Morse function has finite relative indices. The differentials all count some sort of pseudoholomorphic curves.

Lagrangian Intersection Floer homology

Lagrangian Floer homology of two Lagrangian submanifolds of a symplectic manifold is generated by the intersection points of the two submanifolds and its differential counts pseudoholomorphic Whitney discs. It is related to symplectic Floer homology because the graph of a symplectomorphism of a symplectic manifold M is a Lagrangian submanifold of M cross M, and fixed points correspond to intersections of the Lagrangians. It has nice applications to Heegard Floer homology (see below) and in work of Seidel-Smith and Manolescu exhibiting part of the combinatorially defined Khovanov homology as a Lagrangian intersection Floer homology.

Related Topics:
Lagrangian submanifold - Pseudoholomorphic - Whitney discs

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Monumental papers on this subject are due to Fukaya, Oh, Ono, and Ohta, though the recent work on "cluster homology" of Lalonde and Cornea may simplify it.

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• Atiyah-Floer Conjecture