Floer homology

Floer homology is a mathematical tool used in the study of symplectic geometry and low-dimensional topology. First introduced by Andreas Floer in his proof of the Arnold conjecture in symplectic geometry, Floer homology is a novel homology theory arising as an infinite dimensional analog of finite dimensional Morse homology. A similar construction, also introduced by Floer, provides a homology theory associated to three-dimensional manifolds. This theory, along with a number of its generalizations, plays a fundamental role in current investigations into the topology of three- and four-dimensional manifolds. Using techniques from gauge theory, these investigations have provided surprising new insights into the structure of three- and four-dimensional differentiable manifolds.

Floer homology is typically defined by associating an infinite dimensional manifold to the object of interest. In the symplectic version, this is the free loop space of a symplectic manifold, while in the three-dimensional manifold version, it is the space of SU(2)-connections on a three-dimensional manifold. Loosely speaking, Floer homology is the Morse homology computed from a natural function on this infinite dimensional manifold. This function is the symplectic action on the free loop space or the Chern–Simons function on the space of connections. A homology theory is formed from the vector space spanned by the critical points of this function. A linear endomorphism of this vector space is defined by counting the function's gradient flow lines connecting two critical points. Floer homology is then the quotient vector space formed by identifying the image of this endomorphism inside its kernel.

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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 will be 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 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 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 which 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).

The symplectic version of Floer homology figures in a crucial way in the formulation of the homological mirror symmetry conjecture.

Floer homology of three-manifolds

There are several conjecturally equivalent Floer homologies associated to closed three-manifolds. Each yields three types of homology groups, which fit into an exact triangle. Heegaard Floer and Seiberg–Witten Floer homology also yield knot invariants which are formally similar to the combinatorially-defined Khovanov homology. These are closely related to the Donaldson and Seiberg invariants of 4-manifolds, as well as 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 (Juhász 2008) and bordered Floer homology (Lipshitz, Ozsváth & Thurston 2011). These are expected to be 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 beginning with Kronheimer and Mrowka in the Seiberg–Witten case. (A choice of contact structure is required to define embedded contact homology but not the others.)

These theories all come equipped with a priori relative gradings; these have been lifted to absolute gradings (by assigning homotopy classes of 2-plane fields) for SWF, and using the SWF-ECH isomorphism, ECH.

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. 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. 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, also known as monopole Floer homology, is a homology theory of smooth 3-manifolds (equipped with a spinc structure) that is generated by solutions to Seiberg–Witten equations on a 3-manifold and whose differential counts invariant solutions to the Seiberg–Witten equations on the product of a 3-manifold and the real line.

SWF was constructed rigorously in the book of Peter Kronheimer and Tomasz Mrowka. Before that, SWF for rational homology 3-spheres had been constructed by Frøyshov (2002), and certain cases had also been constructed using finite-dimensional approximation in by Kronheimer & Manolescu (2003).

Cliff Taubes has proved that SWF and ECH are isomorphic.

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. Kutluhan, Lee & Taubes (2011) and Colin, Ghiggini & Honda (2011) announced proofs that Heegaard Floer homology is isomorphic to embedded contact homology.

A knot in a three-manifold induces a filtration on the Heegaard Floer homology groups, and the filtered homotopy type is a powerful knot invariant, which categorifies the Alexander polynomial. It was defined by Ozsváth & Szabó (2003) and independently by Rasmussen (2003). It is known to detect knot genus. The knot Floer homology of a knot (as well as the Heegaard Floer homology of the double cover of S^3 branched over a knot) are related by spectral sequences to (variants of) Khovanov homology. (Ozsváth & Szabó 2005)

Using grid diagrams for the Heegaard splittings, Heegaard Floer homology with coefficients in Z_2 and the "HF hat" version of Heegaard Floer homology and the maps induced on it by cobordisms have been given combinatorial constructions by several authors starting out from initial work by Sucharit Sarkar. Similar constructions have yielded combinatorial constructions of knot Floer homology.

Embedded contact homology

Embedded contact homology, due to Michael Hutchings, is an invariant of 3-manifolds (with a distinguished second homology class, corresponding to the choice of a spinc structure in Seiberg–Witten Floer homology, isomorphic (by work of Clifford Taubes) to Seiberg–Witten Floer homology and (by work announced by Kutluhan, Lee & Taubes 2011 and Colin, Ghiggini & Honda 2011) to 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 symplectic 4-manifolds (namely, a contact three-manifold cross R). Its construction is analogous to symplectic field theory, in that it is generated by certain collections of closed Reeb orbits and its differential counts certain holomorphic curves with ends at certain collections of Reeb orbits; 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 which also satisfy a topological condition given by the "ECH index", which in particular implies that the curves considered are (mainly) embedded.

The Weinstein conjecture that a contact 3-manifold has a closed Reeb orbit for any contact form holds on any manifold whose ECH is nontrivial, and was proved by Taubes using techniques closely related to ECH; extensions of this work yielded the isomorphism between ECH and SWF. Many constructions in ECH (including its well-definedness) rely upon this isomorphism.

The contact element of ECH has a particularly nice form: it is the cycle associated to the empty collection of Reeb orbits.

An analog of embedded contact homology may be defined for mapping tori of symplectomorphisms of a surface (possibly with boundary) and is known as periodic Floer homology, generalizing the symplectic Floer homology of surface symplectomorphisms. More generally, it may be defined with respect to any stable Hamiltonian structure on the 3-manifold.

Lagrangian intersection Floer homology

The Lagrangian Floer homology of two Lagrangian submanifolds of a symplectic manifold is the homology of a chain complex which is generated by the intersection points of the two submanifolds and whose differential counts pseudoholomorphic Whitney discs. The symplectic Floer homology of a symplectomorphism of M can be thought of as the special case of Lagrangian Floer homology in which the ambient manifold is M cross M and the Lagrangian submanifolds are the diagonal and the graph of the symplectomorphism. The construction of Heegaard Floer homology (see above) is based on a variant of Lagrangian Floer homology. The theory also appears in work of Seidel–Smith and Manolescu exhibiting what is conjectured to be part of the combinatorially-defined Khovanov homology as a Lagrangian intersection Floer homology.

Given three Lagrangian submanifolds L0, L1, and L2 of a symplectic manifold, there is a product structure on the Lagrangian Floer homology:

HF(L_0, L_1) \otimes HF(L_1,L_2) \rightarrow HF(L_0,L_2),

which is defined by counting holomorphic triangles (that is, holomorphic maps of a triangle whose vertices and edges map to the appropriate intersection points and Lagrangian submanifolds).

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. The Floer homology of a pair of Lagrangian submanifolds may not always exist; when it does, it provides an obstruction to isotoping one Lagrangian away from the other using a Hamiltonian isotopy.

Atiyah–Floer conjecture

The Atiyah–Floer conjecture connects the instanton Floer homology with the Lagrangian intersection Floer homology: Consider a 3-manifold Y with a Heegaard splitting along a surface \Sigma. Then the space of flat connections on \Sigma modulo gauge equivalence is a symplectic manifold of dimension 6g − 6, where g is the genus of the surface \Sigma. In the Heegaard splitting, \Sigma bounds two different 3-manifolds; the space of flat connections modulo gauge equivalence on each 3-manifold with boundary (equivalently, the space of connections on \Sigma that extend over each three manifold) is a Lagrangian submanifold of the space of connections on \Sigma. We may thus consider their Lagrangian intersection Floer homology. Alternately, we can consider the Instanton Floer homology of the 3-manifold Y. The Atiyah–Floer conjecture asserts that these two invariants are isomorphic. Salamon & Wehrheim (2008) are working on a program to prove this conjecture.

Relations to mirror symmetry

The homological mirror symmetry conjecture of Maxim Kontsevich predicts an equality between the Lagrangian Floer homology of Lagrangians in a Calabi–Yau manifold X and the Ext groups of coherent sheaves on the mirror Calabi–Yau manifold. In this situation, one should not focus on the Floer homology groups but on the Floer chain groups. Similar to the pair-of-pants product, one can construct multi-compositions using pseudo-holomorphic n-gons. These compositions satisfy the A_\infty-relations making the category of all (unobstructed) Lagrangian submanifolds in a symplectic manifold into an A_\infty-category, called the Fukaya category.

To be more precise, one must add additional data to the Lagrangian – a grading and a spin structure. A Lagrangian with a choice of these structures is often called a brane in homage to the underlying physics. The Homological Mirror Symmetry conjecture states there is a type of derived Morita equivalence between the Fukaya category of the Calabi–Yau X and a dg category underlying the bounded derived category of coherent sheaves of the mirror, and vice-versa.

Symplectic field theory (SFT)

This is an invariant of contact manifolds and symplectic cobordisms between them, originally due to Yakov Eliashberg, Alexander Givental and Helmut Hofer. The symplectic field theory as well as its subcomplexes, rational symplectic field theory and contact homology, are defined as homologies of differential algebras, which are generated by closed orbits of the Reeb vector field of a chosen contact form. The differential counts certain holomorphic curves in the cylinder over the contact manifold, where the trivial examples are the branched coverings of (trivial) cylinders over closed Reeb orbits. It further includes a linear homology theory, called cylindrical or linearized contact homology (sometimes, by abuse of notation, just contact homology), whose chain groups are vector spaces generated by closed orbits and whose differentials count only holomorphic cylinders. However, cylindrical contact homology is not always defined due to the presence of holomorphic discs. In situations where cylindrical contact homology makes sense, it may be seen as the (slightly modified) "Morse homology" of the action functional on the free loop space which sends a loop to the integral of the contact form alpha over the loop. Reeb orbits are the critical points of this functional.

SFT also associates a relative invariant of a Legendrian submanifold of a contact manifold known as relative contact homology.

In SFT the contact manifolds can be replaced by mapping tori of symplectic manifolds with symplectomorphisms. While the cylindrical contact homology is well-defined and given by the symplectic Floer homologies of powers of the symplectomorphism, (rational) symplectic field theory and contact homology can be considered as generalized symplectic Floer homologies. 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.

Floer homotopy

One conceivable way to construct a Floer homology theory of some object would be to construct a related spectrum whose ordinary homology is the desired Floer homology. Applying other homology theories to such a spectrum could yield other interesting invariants. This strategy was proposed by Ralph Cohen, John Jones, and Graeme Segal, and carried out in certain cases for Seiberg–Witten–Floer homology by Kronheimer & Manolescu (2003) and for the symplectic Floer homology of cotangent bundles by Cohen.

Analytic foundations

Many of these Floer homologies have not been completely and rigorously constructed, and many conjectural equivalences have not been proved. Technical difficulties come up in the analysis involved, especially in constructing compactified moduli spaces of pseudoholomorphic curves. Hofer, in collaboration with Kris Wysocki and Eduard Zehnder, has developed new analytic foundations via their theory of polyfolds and a "general Fredholm theory". While the polyfold project is not yet fully completed, in some important cases transversality was shown using simpler methods.

Computation

Floer homologies are generally difficult to compute explicitly. For instance, the symplectic Floer homology for all surface symplectomorphisms was completed only in 2007. The Heegaard Floer homology has been huge success story in this regard: researchers have exploited its algebraic structure to compute it for various classes of 3-manifolds and indeed found combinatorial algorithms for computation of much of the theory. It is also connected it to existing invariants and structures and many insights into 3-manifold topology have resulted.

References

Books and surveys

Research articles