In mathematics, Floer homology is a tool for studying symplectic geometry and low-dimensional topology. Floer homology is an invariant that arises as an infinite-dimensional analogue of finite-dimensional Morse homology. Andreas Floer introduced the first version of Floer homology, now called symplectic Floer homology, in his 1988 proof of the Arnold conjecture in symplectic geometry. 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. These constructions and their descendants play a fundamental role in current investigations into the topology of symplectic and contact manifolds as well as (smooth) three- and four-dimensional manifolds.
Floer homology is typically defined by associating to the object of interest an infinite-dimensional manifold and a real valued function on it. In the symplectic version, this is the free loop space of a symplectic manifold with the symplectic action functional. 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. Loosely speaking, Floer homology is the Morse homology of the function on the infinite-dimensional manifold. A Floer chain complex is formed from the abelian group spanned by the critical points of the function (or possibly certain collections of critical points). 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). Floer homology is the homology of this chain complex.
The gradient flow line equation, in a situation where Floer's ideas can be successfully applied, is typically a geometrically meaningful and analytically tractable equation. 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. The Gromov compactness theorem is then used to show that the counts of flow lines defining the differential are finite, so that the differential is well-defined and squares to zero. Thus the Floer homology is defined. For instanton Floer homology, the gradient flow equation is exactly the Yang–Mills equation on the three-manifold crossed with the real line.
Symplectic Floer homology
Symplectic Floer Homology (SFH) is a homology theory associated to a symplectic manifold and a nondegenerate symplectomorphism of it. If the symplectomorphism is Hamiltonian, the homology arises from studying the symplectic action functional on the (universal cover of the) free loop space of a symplectic manifold. SFH is invariant under Hamiltonian isotopy of the symplectomorphism.
Here, nondegeneracy means that 1 is not an eigenvalue of the derivative of the symplectomorphism at any of its fixed points. This condition implies that the fixed points are isolated. SFH is the homology of the chain complex generated by the fixed points of such a symplectomorphism, where the differential counts certain pseudoholomorphic curves in the product of the real line and the mapping torus of the symplectomorphism. This itself is a symplectic manifold of dimension two greater than the original manifold. 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. A relative index may be defined between pairs of fixed points, and the differential counts the number of holomorphic cylinders with relative index 1.
The symplectic Floer homology of a Hamiltonian symplectomorphism of a compact manifold is isomorphic to the singular homology of the underlying manifold. Thus, the sum of the Betti numbers of that manifold yields the lower bound predicted by one version of the Arnold conjecture for the number of fixed points for a nondegenerate symplectomorphism. The SFH of a Hamiltonian symplectomorphism also has a pair of pants product that is a deformed cup product equivalent to quantum cohomology. A version of the product also exists for non-exact symplectomorphisms.
For the cotangent bundle of a manifold M, the Floer homology depends on the choice of Hamiltonian due to its noncompactness. For Hamiltonians that are quadratic at infinity, the Floer homology is the singular homology of the free loop space of M (proofs of various versions of this statement are due to Viterbo, Salamon–Weber, Abbondandolo–Schwarz, and Cohen). There are more complicated operations on the Floer homology of a cotangent bundle that correspond to the string topology operations on the homology of the loop space of the underlying manifold.
The symplectic version of Floer homology figures in a crucial way in the formulation of the homological mirror symmetry conjecture.
PSS isomorphism 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:
The Floer cohomology groups of the loop space of a semi-positive symplectic manifold (M,ω) are naturally isomorphic to the ordinary cohomology of M, tensored by a suitable Novikov ring associated the group of covering transformations. This isomorphism intertwines the quantum cup product structure on the cohomology of M with the pair-of-pants product on Floer homology.
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):
for every A in π2(M) where λ≥0 (M is monotone). for every A in 2(M). The minimal Chern Number N ≥ 0 defined by is greater than or equal to n − 2.
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.
This construction of Floer homology explains the independence on the choice of the almost complex structure on M and the isomorphism to Floer homology provided from the ideas of Morse theory and pseudoholomorphic curves, where we must recognize the Poincaré duality between homology and cohomology as the background.
Floer homology of three-manifolds There are several equivalent Floer homologies associated to closed three-manifolds. Each yields three types of homology groups, which fit into an exact triangle. A knot in a three-manifold induces a filtration on the chain complex of each theory, whose chain homotopy type is a knot invariant. (Their homologies satisfy similar formal properties to the combinatorially defined Khovanov homology.)
These homologies are closely related to the Donaldson and Seiberg-Witten invariants of 4-manifolds, as well as to Taubes's Gromov invariant of symplectic 4-manifolds; the differentials of the corresponding three-manifold homologies to these theories are studied by considering solutions to the relevant differential equations (Yang–Mills, Seiberg–Witten, and Cauchy–Riemann, respectively) on the 3-manifold cross R. The 3-manifold Floer homologies should also be the targets of relative invariants for four-manifolds with boundary, related by gluing constructions to the invariants of a closed 4-manifold obtained by gluing together bounded 3-manifolds along their boundaries. (This is closely related to the notion of a topological quantum field theory.) For Heegaard Floer homology, the 3-manifold homology was defined first, and an invariant for closed 4-manifolds was later defined in terms of it.
There are also extensions of the 3-manifold homologies to 3-manifolds with boundary: sutured Floer homology and bordered Floer homology . These are related to the invariants for closed 3-manifolds by gluing formulas for the Floer homology of a 3-manifold described as the union along the boundary of two 3-manifolds with boundary.
The three-manifold Floer homologies also come equipped with a distinguished element of the homology if the three-manifold is equipped with a contact structure. Kronheimer and Mrowka first introduced the contact element in the Seiberg–Witten case. Ozsvath and Szabo constructed it for Heegaard Floer homology using Giroux's relation between contact manifolds and open book decompositions, and it comes for free, as the homology class of the empty set, in embedded contact homology. (Which, unlike the other three, requires a contact structure for its definition. For embedded contact homology see .
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.
Instanton Floer homology This is a three-manifold invariant connected to Donaldson theory introduced by Floer himself. 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). Its critical points are flat connections and its flow lines are instantons, i.e. anti-self-dual connections on the three-manifold crossed with the real line. Instanton Floer homology may be viewed as a generalization of the Casson invariant because the Euler characteristic of the Floer homology agrees with the Casson invariant.
Soon after Floer's introduction of Floer homology, Donaldson realized that cobordisms induce maps. This was the first instance of the structure that came to be known as a topological quantum field theory.
Seiberg–Witten Floer homology Seiberg–Witten Floer homology or monopole Floer homology is a homology theory for smooth 3-manifolds (equipped with a spinc structure). It may be viewed as the Morse homology of the Chern–Simons–Dirac functional on U(1) connections on the three-manifold. The associated gradient flow equation corresponds to the Seiberg–Witten equations on the 3-manifold crossed with the real line. Equivalently, the generators of the chain complex are translation-invariant solutions to Seiberg–Witten equations (known as monopoles) on the product of a 3-manifold and the real line, and the differential counts solutions to the Seiberg–Witten equations on the product of a three-manifold and the real line, which are asymptotic to invariant solutions at infinity and negative infinity.
One version of Seiberg–Witten–Floer homology was constructed rigorously in the monograph Monopoles and Three-manifolds by Peter Kronheimer and Tomasz Mrowka, where it is known as monopole Floer homology. Taubes has shown that it is isomorphic to embedded contact homology. Alternate constructions of SWF for rational homology 3-spheres have been given by and ; they are known to agree.
Heegaard Floer homology 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. announced a proof that Heegaard Floer homology is isomorphic to Seiberg–Witten Floer homology, and announced a proof that the plus-version of Heegaard Floer homology (with reverse orientation) is isomorphic to embedded contact homology.