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.

Symplectic Floer homology

Symplectic Floer homology is a homology theory associated to a symplectic manifold and a nondegenerate symplectomorphism of it. It is generated by fixed points of the symplectomorphism, and counts pseudoholomorphic curves in R cross the mapping torus of the symplectomorphism. It is invariant under Hamiltonian isotopy of the symplectomorphism.

Related Topics:
Symplectic manifold - Symplectomorphism - Fixed points - Pseudoholomorphic curve - Mapping torus - Hamiltonian isotopy

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The symplectic Floer homology of an exact symplectomorphism--i.e. one that is a Hamiltonian deformation of the identity--is isomorphic to the singular homology of the underlying manifold. Thus, the Betti numbers of that manifold yield the lower bounds predicted in the Arnold Conjectures for the number of fixed points for a nondegenerate symplectomorphism. The SFH of an exact symplectomorphism also

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has a pair of pants product which is a deformed cup product equivalent to quantum cohomology. A version of the product also exists for non-exact symplectomorphisms

Related Topics:
Pair of pants - Cup product - Quantum cohomology

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