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The several conjecturally equivalent Floer homologies of Three-manifolds all yield three types of homology groups, that fit into an Exact Triangle . This is formally similar to the combinatorially-defined Khovanov Homology , which is known to be related by a Spectral Sequence to Heegaard Floer Homology . The Three-manifold theories also come equipped with a distinguished element if the Three-manifold is equipped with a Contact Structure (A contact structure is required to define Embedded Contact Homology but not the others). They should also have corresponding relative invariants for four-manifolds with boundary values in the Floer homologies of the boundaries. This last is closely related to the notion of a Topological Quantum Field Theory . SYMPLECTIC FLOER HOMOLOGY Symplectic Floer homology is a homology theory associated to a Symplectic Manifold and a nondegenerate Symplectomorphism of it. It is generated by Fixed Points of the symplectomorphism, and counts Pseudoholomorphic Curve s in the product of the real line and the Mapping Torus of the symplectomorphism. It is invariant under Hamiltonian Isotopy of the symplectomorphism. It comes from studying the Symplectic Action functional on the (universal cover of the) loop space of a symplectic manifold. The symplectic Floer homology of an exact symplectomorphism--i.e. one that is a Hamiltonian deformation of the identity--is isomorphic to the singular homology of the underlying manifold. Thus, the Betti numbers of that manifold yield the lower bounds predicted in the Arnold Conjecture s for the number of fixed points for a nondegenerate symplectomorphism. The SFH of an exact 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. INSTANTON FLOER HOMOLOGY This is a three-manifold invariant connected to Donaldson Theory . It studies the Chern-Simons functional on the space of Connection s on an SU(2) -bundle over a manifold. Its critical points are Flat Connection s and its flow lines are Instanton s. LAGRANGIAN INTERSECTION FLOER HOMOLOGY Lagrangian Floer homology of two Lagrangian Submanifold s of a symplectic manifold is generated by the intersection points of the two submanifolds and its differential counts Pseudoholomorphic Whitney Discs . It is related to symplectic Floer homology because the graph of a symplectomorphism of a symplectic manifold M is a Lagrangian submanifold of M cross M, and fixed points correspond to intersections of the Lagrangians. It has nice applications to Heegard Floer homology (see below) and in work of Seidel-Smith and Manolescu exhibiting 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: :, 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). Difficult papers on this subject are due to Fukaya, Oh, Ono, and Ohta; the recent work on " Cluster Homology " of Lalonde and Cornea may simplify 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 . Then the space of Flat Connection s on modulo Gauge Equivalence is a symplectic manifold of dimension ''6g - 6'', where ''g'' is the Genus of the surface . In the Heegard splitting, 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 that extend over each three manifold) is a Lagrangian submanifold of the space of connections on . 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. Katrin Wehrheim and Dietmar Salamon are working on a program to prove this conjecture. SEIBERG-WITTEN FLOER HOMOLOGY Seiberg-Witten Floer homology, also known as monopole Floer homology, is a homology theory of smooth 3-manifold s (equipped with a Spin''c'' 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 is constructed rigorously in certain cases using Finite-dimensional Approximation in papers by Ciprian Manolescu , and Manolescu with Peter Kronheimer ; a more traditional approach is taken in the forthcoming book of Kronheimer and Tomasz Mrowka . HEEGAARD FLOER HOMOLOGY Heegaard Floer homology is an invariant of a closed 3-manifold equipped with a spin''c'' structure. It is computed using a Heegaard Diagram of the space via Lagrangian Floer homology. It is conjecturally equivalent to Seiberg-Witten-Floer homology. A knot in a three-manifold induces a filtration on the homology groups, and the filtered homotopy type is a powerful Knot Invariant , which Categorifies the Alexander Polynomial . It was defined and developed in a long series of papers by Peter Ozsvath and Zoltan Szabo ; the associated knot invariant was independently discovered by Jacob Rasmussen . SYMPLECTIC FIELD THEORY This is an invariant of Contact Manifold s and symplectic Cobordism s between them, originally due to Yasha Eliashberg , Alexander Givental and Helmut Hofer . It includes a homology theory, often called contact homology, whose chains are generated by closed orbits of the Reeb Vector Field and whose differentials count punctured holomorphic curves in the symplectization of a contact manifold. Contact homology is associated to the functional on the loop space given by integrating a contact one-form around a loop; the critical points are precisely the closed Reeb orbits. The entire symplectic field theory takes the form of a Differential Graded Algebra . SFT also associates a relative invariant of a Legendrian Submanifold of a contact manifold known as Relative Contact Homology . EMBEDDED CONTACT HOMOLOGY Embedded contact homology, due to Michael Hutchings and Michael Sullivan , is an invariant of 3-manifolds (with a distinguished second homology class, analogous to the choice of a spin''c'' structure in Seiberg-Witten Floer homology) 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-manifold s 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 equivalently HFH or SWF) is nontrivial. Embedded contact homology is closely related to the periodic Floer homology defined by Hutchings and Michael Thaddeus. 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 Space s of pseudoholomorphic curves. Hofer, in collaboration with Kris Wysocki and Eduard Zehnder, has developed new analytic foundations via their theory of Polyfold s. A preliminary version of the first volume (of four) of their book on their theory was circulated in 2005. 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. REFERENCES
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