In mathematics, Floer homology is a tool for studying symplectic geometry and low-dimensional topology.
Floer also developed a closely related theory for Lagrangian submanifolds of a symplectic manifold.
A third construction, also due to Floer, associates homology groups to closed three-dimensional manifolds using the Yang–Mills functional.
Floer homology is typically defined by associating to the object of interest an infinite-dimensional manifold and a real valued function on it.
For the (instanton) version for three-manifolds, it is the space of SU(2)-connections on a three-dimensional manifold with the Chern–Simons functional.
The differential of the chain complex is defined by counting the flow lines of the function's gradient vector field connecting fixed pairs of critical points (or collections thereof).
For symplectic Floer homology, the gradient flow equation for a path in the loopspace is (a perturbed version of) the Cauchy–Riemann equation for a map of a cylinder (the total space of the path of loops) to the symplectic manifold of interest; solutions are known as pseudoholomorphic curves.
For an appropriate choice of almost complex structure, punctured holomorphic curves (of finite energy) in it have cylindrical ends asymptotic to the loops in the mapping torus corresponding to fixed points of the symplectomorphism.
For the cotangent bundle of a manifold M, the Floer homology depends on the choice of Hamiltonian due to its noncompactness.
In 1996 S. Piunikhin, D. Salamon and M. Schwarz summarized the results about the relation between Floer homology and quantum cohomology and formulated as the following.Piunikhin, Salamon & Schwarz (1996) The above condition of semi-positive and the compactness of symplectic manifold M is required for us to obtain Novikov ring and for the definition of both Floer homology and quantum cohomology.
The semi-positive condition means that one of the following holds (note that the three cases are not disjoint): The quantum cohomology group of symplectic manifold M can be defined as the tensor products of the ordinary cohomology with Novikov ring Λ, i.e.
These theories all come equipped with a priori relative gradings; these have been lifted to absolute gradings (by homotopy classes of oriented 2-plane fields) by Kronheimer and Mrowka (for SWF), Gripp and Huang (for HF), and Hutchings (for ECH).
Cristofaro-Gardiner has shown that Taubes' isomorphism between ECH and Seiberg–Witten Floer cohomology preserves these absolute gradings.
It is obtained using the Chern–Simons functional on the space of connections on a principal SU(2)-bundle over the three-manifold (more precisely, homology 3-spheres).
Heegaard Floer homology // ⓘ is an invariant due to Peter Ozsváth and Zoltán Szabó of a closed 3-manifold equipped with a spinc structure.
It is computed using a Heegaard diagram of the space via a construction analogous to Lagrangian Floer homology.
Using grid diagrams for the Heegaard splittings, knot Floer homology was given a combinatorial construction by Manolescu, Ozsváth & Sarkar (2009).
The "plus" and "minus" versions of Heegaard Floer homology, and the related Ozsváth–Szabó four-manifold invariants, can be described combinatorially as well (Manolescu, Ozsváth & Thurston 2009).
It differs from SFT in technical conditions on the collections of Reeb orbits that generate it—and in not counting all holomorphic curves with Fredholm index 1 with given ends, but only those that also satisfy a topological condition given by the ECH index, which in particular implies that the curves considered are (mainly) embedded.
The contact element of ECH has a particularly nice form: it is the cycle associated to the empty collection of Reeb orbits.
The Lagrangian Floer homology of two transversely intersecting Lagrangian submanifolds of a symplectic manifold is the homology of a chain complex generated by the intersection points of the two submanifolds and whose differential counts pseudoholomorphic Whitney discs.
Papers on this subject are due to Fukaya, Oh, Ono, and Ohta; the recent work on "cluster homology" of Lalonde and Cornea offer a different approach to it.
bounds two different 3-manifolds; the space of flat connections modulo gauge equivalence on each 3-manifold with boundary embeds into
The Homological Mirror Symmetry conjecture states there is a type of derived Morita equivalence between the Fukaya category of the Calabi–Yau
This is an invariant of contact manifolds and symplectic cobordisms between them, originally due to Yakov Eliashberg, Alexander Givental and Helmut Hofer.
However, cylindrical contact homology is not always defined due to the presence of holomorphic discs and a lack of regularity and transversality results.
In the important case when the symplectomorphism is the time-one map of a time-dependent Hamiltonian, it was however shown that these higher invariants do not contain any further information.
This approach was the basis of Manolescu's 2013 construction of Pin (2)-equivariant Seiberg–Witten Floer homology, with which he disproved the Triangulation Conjecture for manifolds of dimension 5 and higher.
Technical difficulties come up in the analysis involved, especially in constructing compactified moduli spaces of pseudoholomorphic curves.
While the polyfold project is not yet fully completed, in some important cases transversality was shown using simpler methods.