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.

Computation

Floer homologies are generally difficult to compute explicitly. For instance, the symplectic Floer homology is not even known for all surface symplectomorphisms. The Heegard Floer homology has been something of a success story in this regard: researchers have exploited its algebraic structure to compute it for various classes of 3-manifolds and connected it to existing invariants and structures; some insights into 3-manifold topology have resulted.

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