Floer homology

Floer homology ( FH) referred to in the topology and differential geometry, a group of similarly designed homology invariants. They have their origin in the work of Andreas Floer and have since been constantly evolving. Floer extended the Morse homology ( Morse theory ) finite-dimensional manifolds in cases where has the Morse function is no longer finite, but only " relatively finite " indices, especially in symplectic manifolds, where the " differentials " of homology construction pseudoholomorphic curves count.

  • 3.1 The Atiyah - Floer conjecture
  • 3.2 Connection to the mirror symmetry

Symplectic Floer homology ( SFH )

In this case, the Floer homology is a symplectic manifold (such as the phase spaces of classical mechanics ) M with a scale on their non-degenerate Symplektomorphismus ( symplectic mapping or "river", he gets the volume) F connected. If this is the Hamilton - type, can be defined on the space of closed paths of M (loop space) a functional effect (action functional), and the SFH results from the study of this functional. SFH is invariant under a Hamiltonian isotopy of F.

" Non- degenerate" means that the eigenvalues ​​of the derivative dF of the flux F at its fixing points are not equal to 1, that the fixed points are isolated points.

The SFH is then as homology ( group) of the fixed points defined by this chain complex ( chain complex) defined. The " differential " in this chain complex ( " differential " in the sense of algebraic topology, as in the following chapters) counts certain pseudoholomorphic curves in the product R x T, where T is called the mapping torus of F. T is itself a symplectic manifold with a by 2 major dimension as M. For a suitable choice of pseudoholomorphen structure punctured holomorphic curves in T asymptotic cylindrical ends which correspond to the fixed points of F. The central idea of Floer it to define a relative index between pairs of fixed points was, and the " differential " counts the number of holomorphic cylinders with relative index 1

The SFH of a Hamiltonian Symplektomorphismus F is isomorphic to the singular homology of the underlying manifold M. Therefore, the sums of the Betti numbers of M provide a lower limit for the number of fixed points of a non-degenerate Symplektomorphismus F ( Arnold conjecture ). The SFH of a Hamiltonian Symplektomorphismus F also have a "pair of pants" product, which is a deformed cup product equivalent to the quantum cohomology.

Floer homology of 3-manifolds

The various (presumably equivalent ) Floer homology of three-dimensional manifolds deliver homology groups forming an exact triangle (exact triangle ). The Heegaard Floer homology and knot invariants also provides similar formally to the combinatorially defined Khovanov homology.

A special feature of FH of 3-manifolds will be assumed if these manifolds have contact structures, because then can be " embedded contact homology " define.

You should also invariants for 4 -manifolds arise over the Floer homology of the 3 -dimensional edges of these manifolds. Related to this is the concept of topological quantum field theory.

Instanton Floer homology

This is an invariant of 3-manifolds M, which is connected to a theory by Simon Donaldson. It results from the consideration of the Chern - Simons functional on the space of connection forms ( connections ) of the SU (2) - principal bundle over M. Its critical points are flat connections (flat connections ), and its flow lines are instantons ( "anti- self dual connections " on R x M)

Seiberg -Witten Floer homology ( SWF)

Seiberg -Witten Floer homology, also known as monopoly -FH is a homology theory of smooth 3-manifolds, equipped with a Spinc structure, which is given by the solutions of the Seiberg -Witten equations on a 3-manifold M and their " differential " solutions of the Seiberg -Witten equations on the product M × R one.

The exact construction of these homology in some special cases and in finite approximation is done in some works by Ciprian Manolescu and Peter Kronheimer. A more conventional approach is taken in a book by Kronheimer and Tomasz Mrowka.

Heegaard Floer homology

Heegaard Floer homology is an invariant of a closed spinc 3-manifold Y. It is constructed by means of Heegaard decomposition of Y by Lagrangian Floer homology. This gives several homology groups that are by exact sequences relate to each other. Similarly, you can assign each W 4-dimensional cobordism between two 3-manifolds Y and Y 'is a morphism between the Floer homologies. The exact sequences transform naturally under the associated morphisms. By means of constructing suitable filtrations can be constructed invariants. An example of this is associated to a node K in a 3-manifold Y knot homology. Another example is the so-called contact class, an invariant of contact structures.

The Heegaard Floer homology has been developed in a long series of works by Peter Ozsváth and Zoltán Szabó, the associated knot invariant was also discovered independently by Jacob Rasmussen.

Embedded contact homology (embedded contact homology, ECH)

It was replaced by Michael Hutchings and Michael Sullivan introduced an invariant of 3-manifolds with an additional defined second homology class (analogous to the spin- c structure in Seiberg -Witten FH). It is believed that it is equivalent to Seiberg -Witten and Heegaard FH -FH. It can be considered an extension of Taubes ' Gromov invariant, which is known that it is the Seiberg - Witten invariant equivalent, and is an invariant of mappings of closed symplectic 4 -manifolds to certain non - compact 4- manifolds.

The design of these FH is analogous to the symplectic field theory, but only refers embedded pseudoholomorphic curves a ( with a few additional technical conditions ). For manifolds with non-trivial ECH there is a conjecture of Weinstein, who was with techniques that are closely related to ECH, proved by Taubes.

Lagrange cut -FH ( Lagrangian Floer homology intersection )

The Lagrange -FH two Lagrangian submanifolds of a symplectic manifold M is generated by the intersections of the two submanifolds. Your " differential " counts pseudoholomorphic Whitney disks. It is connected to the SFH, since the graph of Symplektomorphismus M is a Lagrange submanifold of M x M, and its fixed points corresponding to the sections of the Lagrangemannigfaltigkeit. She has nice applications in the Heegaard -FH (see below) and expressed in the work of Seidel -Smith and Manolescu, the parts of the combinatorially defined Khovanov homology as a Lagrange cut -FH.

There are three Lagrangian submanifolds L0, L1 and L2 given a symplectic manifold. Then there is a product structure on the Lagrangian FH:

By counting holomorphic triangles ( ie holomorphic maps of triangles whose vertices and edges are mapped to the corresponding intersection points and Lagrangian submanifolds ) is defined.

Work on this is by Kenji Fukaya, Y. Oh, Kaoru Ono, and H. Ohta; or any other access to the work on the "cluster homology" of François Lalonde and Octav Cornea. The FH of pairs of Lagrangian submanifolds need not always exist, but if it exists, it provides an obstruction for the " Isotopierung " of a submanifold of the other by means of Hamiltonian isotopies.

The Atiyah - Floer conjecture

The Atiyah - Floer conjecture connects the instanton Floer homology with Lagrangian intersection Floer homology: Let M be a 3-manifold with a Heegaard fragmentation along a surface. Then the space of the " flat bundle " (flat connections, ie vanishing connection form) modulo gauge transformations on a symplectic manifold of dimension ( 6g - 6), where g is the genus of the surface.

In the Heegard - bounded fragmentation two different 3-manifolds; the space of flat bundles modulo gauge transformations on each 3- manifold with boundary ( or equivalently, the space of connection forms on the leaves continue on each of the two 3-manifolds ) is a Lagrangian submanifold of the space of connection forms ( connections ) on. One can therefore consider its Lagrangian intersection Floer homology or alternatively, the instanton Floer homology of the 3-manifold M. The Atiyah - Floer conjecture states that Isormophie these two invariants. Katrin Wehrheim and Dietmar Salamon working on a program to prove this conjecture.

Connections to the mirror symmetry

The homological mirror symmetry conjecture ( mirror symmetry) of Maxim Konze Malevich states the equivalence of the Lagrangian FH of Lagrangian submanifolds in Calabi -Yau manifolds X and Ext- groups of coherent sheaves on the mirror Calabi -Yau manifold advance. More interesting than the FH- groups here are the Floer chain groups (chain groups). Similar to the " pair -of- pants product " can -Gone form of juxtapositions pseudoholomorpher curves. These structures satisfy the relations, thus making the category of all Lagrangian submanifolds (without obstructions ) in a symplectic manifold to a category, called the Fukaya category.

More specifically, additional structures must be added to the Lagrangemannigfaltigkeit, namely a gradation and a spin structure (analogous to the physics called " brane "). Then the conjecture states that a derived - Morita equivalence between the Fukaya category of Calabi -Yau spaces and the dg - category of the derived category ( derived category ) consists of coherent sheaves on the mirror manifold (and vice versa).

Symplectic field theory (SFT )

This is an invariant of contact manifolds (more generally manifolds with a stable Hamiltonian structure ) and the symplectic Kobordismen between them. She is originally from Yakov Eliaschberg, Alexander Givental, and Helmut Hofer. She is - as well as their sub- complexes, the rational symplectic field theory and contact homology as defined homology of Differentialalgebren generated by closed orbits of Reeb vector fields of a contact form. The " differential " counts here certain holomorphic curves in the cylinder R x M via the contact manifold M, the trivial examples are the branched superpositions of ( trivial ) cylinders over closed Reeb orbits. There is a linear homology theory, called cylindrical or linearized contact homology whose chain groups are the vector spaces generated by closed orbits and their differentials are only holomorphic cylinder. Due to the existence of holomorphic disks, the cylindrical contact homology is not always defined. If it is defined, it can as a ( slightly modified ) " Morse homology" of the action functionals are seen on the loop space, which assigns a loop, the integral of a contact form on this loop. " Reeb orbits " are the critical points of this functional.

SFT also associated a relative invariant of a Legendre submanifold of a contact manifold, the "relative contact homology ".

In SFT the contact manifolds can imaging operators ( mapping tori ) of symplectic manifolds are replaced with symplectomorphisms. While the cylindrical contact homology is well-defined ( and is given by the SFH of the powers of symplectomorphisms ), symplectic field theory and contact homology can (rational ) can be regarded as a generalized SFH.

Similarly, an analogue of the " embedded contact homology" (ECH ) for the imaging gates of symplectomorphisms of a surface can be defined ( with border ), the " Periodic FH", which generalizes the SFH of surface symplectomorphisms. It is probably connected with the ECH.

Floer homotopy

One possible way of FH theory to construct an object, would be the construction of an associated " spectrum" whose ordinary homology would be looking for FH. Other invariants would result from the application of other homology theories on this spectrum. The strategy was proposed by Ralph Cohen, John Jones, and Graeme Segal and carried out in certain cases for the Seiberg -Witten FH of Kronheimer and Manolescu and for the symplectic FH of cotangent bundles of Cohen.

Development of techniques

Many of these FH are not complete and was strictly designed, and many suspected equivalences are still open. Problems arise from technical difficulties such as in the compactification of the moduli spaces of curves pseudoholomorphen. Hofer has joined forces with Kris Wysocki and Eduard Zehnder new techniques with their theories of Polyfaltigkeiten and the " generalized Fredholm theory."

Calculation

Floer homology ( FH) are generally difficult to compute explicitly. For example, the symplectic FH is not even known for all surface symplectomorphisms. The Heegaard FH is the exception - it has been calculated for various classes of 3 -manifolds, and while their relationship was illuminated with other invariants.

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