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.

Embedded Contact Homology

Embedded contact homology, due to Michael Hutchings and Michael Sullivan, is an invariant of 3-manifolds conjecturally equivalent to Seiberg-Witten and Heegaard Floer homology. It may be seen as an extension of Taubes's Gromov Invariant, known to be equivalent to the Seiberg-Witten invariant, from closed symplectic 4-manifolds to certain non-compact 4-manifolds. Its construction is analogous to Symplectic Field theory, but it considers only embedded pseudoholomorphic curves satisfying a few technical conditions. The Weinstein Conjecture holds on any manifold whose ECH (or equivallently HFH or SWF) is nontrivial.

Related Topics:
Michael Hutchings - Michael Sullivan - Taubes's Gromov Invariant - Seiberg-Witten invariant - 4-manifold - Weinstein Conjecture

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Embedded contact homology is closely related to the periodic Floer homology defined by Hutchings and Michael Thaddeus.

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