Floer homology

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In mathematics, Floer homology is 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.

The several conjecturally equivalent Floer homologies of three-manifolds all yield three types of homology groups, which fit into an exact triangle. Heegard Floer homology yields a knot invariant, which is formally similar to the combinatorially-defined Khovanov homology. A variant of Khovanov homology is known to be related by a spectral sequence to Heegaard Floer homology of a double cover branched along a knot. (Ozsvath-Szabo 2005). 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.

1 Symplectic Floer homology
2 Instanton Floer homology
3 Lagrangian intersection Floer homology
- 3.1 Atiyah-Floer conjecture
- 3.2 Relations to Mirror symmetry
4 Seiberg-Witten Floer homology
5 Heegaard Floer homology
6 Symplectic field theory
7 Embedded contact homology
8 Analytic foundations
9 Computation
10 References

[edit] Symplectic Floer homology

Symplectic Floer homology 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) 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 is 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.

[edit] Instanton Floer homology

This is a three-manifold invariant connected to Donaldson theory. 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.

[edit] Lagrangian intersection Floer homology

Lagrangian Floer homology of two Lagrangian submanifolds 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:

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

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.

[edit] 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 $Σ$ . Then the space of flat connections on $Σ$ modulo gauge equivalence is a symplectic manifold of dimension 6g - 6, where g is the genus of the surface $Σ$ . In the Heegard splitting, $Σ$ 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 $Σ$ that extend over each three manifold) is a Lagrangian submanifold of the space of connections on $Σ$ . 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.

[edit] 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.

[edit] 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 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.

[edit] 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 Ozsváth and Zoltan Szabó; the associated knot invariant was independently discovered by Jacob Rasmussen.

[edit] Symplectic field theory

This is an invariant of contact manifolds and symplectic cobordisms between them, originally due to Yasha 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 and the differential counts certain holomorphic curves in the cylinder over the contact manifold. 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 "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.

In the general geometric setup of 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.

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

[edit] 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-manifolds 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.

[edit] 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. A preliminary version of the first volume (of four) of their book on their theory was circulated in 2005.

[edit] 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.

[edit] References

Books and Surveys

Atiyah, Michael New invariants of 3- and 4-dimensional manifolds. The mathematical heritage of Hermann Weyl (Durham, NC, 1987), 285-299, Proc. Sympos. Pure Math., 48, Amer. Math. Soc., Providence, RI, 1988.
David A. Ellwood (Editor), Peter S. Ozsvath (Editor), Andras I. Stipsicz (Editor), Zoltan Szabo (Editor).(2006). Floer Homology, Gauge Theory, And Low-dimensional Topology Proceedings of the Clay Mathematics Institute 2004 Summer School, Alfred Renyi Institute of ... 2004 (Clay Mathematics Proceedings, V. 5) (Paperback), ISBN 0-8218-3845-8
McDuff, Dusa & Salamon, Dietmar. (1998). Introduction to Symplectic Topology. Oxford Mathematical Monographs, ISBN 0-19-850451-9. Floer homology in this book

McDuff, Dusa. (2005). Floer theory and low dimensional topology. To be published in Notices of AMS, 2006. Preprint from McDuff's homepage

Banyaga, Augustin & Hurtubise, David. (2004). Lectures on Morse Homology. Dordrecht: Kluwer Academic Publishers. ISBN 1-4020-2695-1. Floer homology in this book

Donaldson, Simon K.; with the assistance of M. Furuta and D. Kotschick. (2002). Homology groups in Yang-Mills theory. (Cambridge tracts in mathematics; 147) Cambridge: University Press. ISBN 0-521-80803-0.

Schwarz, Matthias. Morse H.omology

- 1993 - Birkhäuser

Research Articles

Ozsváth, Peter and Szabó, Zoltán. On the Heegaard Floer homology of branched double-covers. Adv. Math. 194 (2005), no. 1, 1--33. Also available as a preprint.

Floer, Andreas, The unregularized gradient flow of the symplectic action. Comm. Pure Appl. Math. 41 (1988), 775-813.

Floer, Andreas. An instanton-invariant for 3-manifolds. Comm. Math. Phys. 118 (1988), no. 2, 215–240. Project Euclid

Floer, Andreas. Morse theory for Lagrangian intersections. J. Differential Geom. 28 (1988), no. 3, 513–547.

Floer, Andreas. Cuplength estimates on Lagrangian intersections. Comm. Pure Appl. Math. 42 (1989), no. 4, 335–356.

Gromov, Mikhail, Pseudo holomorphic curves in symplectic manifolds. Inventiones Mathematicae (1985), vol. 82 number 2, pp. 307-347.

Floer, Andreas, Symplectic fixed points and holomorphic spheres. Comm. Math. Phys. 120, no. 4 (1989), 575–611.

Floer, Andreas, “Witten's complex and infinite dimensional Morse Theory”. J. Diff. Geom. 30 (1989), p. 202-221.

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