Floer homology

Floer homologies (FH) denotes a group of similarly constructed homology invariants in topology and differential geometry . They have their origins in the work of Andreas Floer and have been continuously developed since then. Floer extended the Morse homology ( Morse theory ) of finite-dimensional manifolds to cases in which the Morse function no longer has finite but only "relatively finite" indices, especially in symplectic manifolds, where the "differentials" of the homology construction Count pseudoholomorphic curves .

Symplectic Floer Homology (SFH)

In this case, the Floer homology is defined for a symplectic manifold (like the phase spaces of classical mechanics) with a non-degenerate symplectomorphism operating on it (symplectic mapping, in particular it contains the volume) . “Not degenerate” means that the eigenvalues of the derivative in the fixed points are different from all of 1, so the fixed points are isolated points . ${\ displaystyle M}$ ${\ displaystyle F}$ ${\ displaystyle dF}$ ${\ displaystyle F}$

If defined by a Hamiltonian flow , an action functional can be defined on the closed path space of (loop space), and the SFH results from the study of this functional. SFH is invariant under a Hamiltonian isotopy of . ${\ displaystyle F}$ ${\ displaystyle \ Omega M}$ ${\ displaystyle M}$ ${\ displaystyle F}$

The SFH is then used as homology defined by these fixed points of the chain complex defined (chain complex). The “differential” in this chain complex (“differential” in the sense of algebraic topology, also in the following chapters) counts certain pseudo-alcoholomorphic curves in the product , the so-called mapping torus being of . is itself a symplectic manifold with a dimension 2 larger than . For a suitable choice of the almost-complex structure dotted have pseudo-holomorphic curves in asymptotically cylindrical ends, the fixing points of the sector. Floer's central idea was to define a relative index between pairs of fixed points, and the "differential" counts the number of pseudo-holomorphic cylinders with relative index 1. ${\ displaystyle \ mathbb {R} \ times T}$ ${\ displaystyle T}$ ${\ displaystyle F}$ ${\ displaystyle T}$ ${\ displaystyle M}$ ${\ displaystyle \ mathbb {R} \ times T}$ ${\ displaystyle F}$

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

Floer homology of 3-manifolds

The different (presumably equivalent) Floer homologies for three-dimensional manifolds yield homology groups that form an exact triangle . The Heegaard-Floer homology also provides knot invariants and is formally similar to the combinatorially defined Khovanov homology .

A specialty of the FH of 3-manifolds occurs if these manifolds have contact structures, because then "embedded contact homologies" can be defined.

So for invariants for 4-manifolds, too, the Floer homologies of the 3-dimensional edges of these manifolds should result. Linked to this is the concept of topological quantum field theory .

Instanton-Floer homology

This is a 3-manifold invariant associated with a theory by Simon Donaldson . It results from the consideration of the Chern-Simons functional on the space of related forms (connections) of the SU (2) - the main fiber bundle over . Its critical points are flat connections , and its flow lines are instantons ("anti-self dual connections" on ) ${\ displaystyle M}$ ${\ displaystyle M}$ ${\ displaystyle \ mathbb {R} \ times M}$

Seiberg-Witten-Floer homology (SWF)

Seiberg-Witten-Floer homology , also known as monopoly FH, is a homology theory of smooth 3-manifolds, provided with a spin ^c structure, whose chain complex is given by the solutions of the Seiberg-Witten equations on a 3-manifold and whose "differential" solutions of the Seiberg-Witten equations count on the product . ${\ displaystyle M}$ ${\ displaystyle M \ times \ mathbb {R}}$

The exact construction of this homology in some special cases and in finite-dimensional approximation is carried out in some works by Ciprian Manolescu and Peter Kronheimer . A more conventional route is taken in a book by Kronheimer and Tomasz Mrowka .

Heegaard-Floer homology

Heegaard-Floer homology is an invariant of a closed spin ^c -3 manifold . It is constructed using the Heegaard decomposition of Lagrange-Floer homology. Several homology groups are obtained, which are related to one another through exact sequences. In a similar way, one can assign each 4-dimensional cobordism between two 3-manifolds and a morphism between the Floer homologies. The exact sequences naturally transform among the associated morphisms. By constructing suitable filters , invariants can be constructed. An example of this is the knot homology associated with a knot in a 3-manifold . Another example is the so-called contact homology, an invariant of contact structures. ${\ displaystyle Y}$ ${\ displaystyle Y}$ ${\ displaystyle W}$ ${\ displaystyle Y}$ ${\ displaystyle Y ^ {\ prime}}$ ${\ displaystyle K}$ ${\ displaystyle Y}$

The Heegaard-Floer homology was 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 (ECH)

It was introduced by Michael Hutchings and Michael Sullivan as an invariant of 3-manifolds with an additionally defined 2nd homology class (analogous to the spin-c structure in Seiberg-Witten-FH). It is believed that it is equivalent to Seiberg-Witten-FH and Heegaard-FH. It can be understood as an extension of Taubes' Gromov invariant, which is known to be equivalent to the Seiberg-Witten invariant, and which is an invariant of mappings from closed symplectic 4-manifolds to certain non-compact 4-manifolds.

The construction of this FH is analogous to the symplectic field theory, but only includes embedded pseudoholomorphic curves (with a few additional technical conditions). For manifolds with non-trivial ECH, there is a conjecture by Weinstein, which was proved by Taubes using techniques closely related to ECH.

Lagrangian intersection Floer homology

The Lagrangian FH of two Lagrangian submanifolds of a symplectic manifold is generated by the intersection of the two submanifolds. Their “differential” counts pseudo-alcoholomorphic Whitney disks . It is connected to the SFH, since the graph of a symplectomorphism of is a Lagrangian submanifold of , and the fixed points correspond to the intersections of the graph with the diagonal, which is also a Lagrangian submanifold. It has nice applications in the Heegaard UAS (see below) and in the work of Seidel- Smith and Manolescu, which express parts of the combinatorially defined Khovanov homology as a Lagrange cut UAS. ${\ displaystyle M}$ ${\ displaystyle M}$ ${\ displaystyle M \ times M}$

Let there be three Lagrange submanifolds and one symplectic manifold. Then there is a product structure on the Lagrange-FH: ${\ displaystyle L_ {0}, L_ {1}}$ ${\ displaystyle L_ {2}}$

{\ displaystyle HF (L_ {0}, L_ {1}) \ otimes HF (L_ {1}, L_ {2}) \ rightarrow HF (L_ {0}, L_ {2})}

,

which is defined by counting pseudo-holomorphic triangles (i.e. pseudo-holomorphic mappings of triangles whose vertices and edges are mapped to the corresponding intersection points and Lagrange submanifolds).

Works on this are by Kenji Fukaya , Y. Oh, Kaoru Ono , and H. Ohta; or in another approach in the work on “ cluster homology ” by François Lalonde and Octav Cornea. The FH of pairs of Lagrange submanifolds need not always exist, but when it does exist it provides an obstruction to an " isotopy " of one submanifold into the other by means of a Hamiltonian isotope.

The Atiyah Floer Conjecture

The Atiyah-Floer conjecture combines the Instanton-Floer homology with the Lagrange cut-Floer homology: Let M be a 3-manifold with a Heegaard cut along a surface . Then the space of the “ flat bundles ” (flat connections, i.e. vanishing curvature form ) on modulo gauge transformations is a symplectic manifold of the dimension , where the gender of the surface is. ${\ displaystyle \ Sigma}$ ${\ displaystyle \ Sigma}$ ${\ displaystyle 6g-6}$ ${\ displaystyle g}$ ${\ displaystyle \ Sigma}$

In the Heegard cut, two handle bodies bordered ; the space of the flat bundles modulo gauge transformations on every 3- manifold with a boundary (or, equivalently, the space of the connected forms to which each of the two 3-manifolds can be extended) is a Lagrangian submanifold of the space of the connected forms (connections) . So one can consider their Lagrange-Cut-Floer homology or, alternatively, the Instanton-Floer homology of the 3-manifold M. The Atiyah-Floer conjecture states the isomorphism of these two invariants. Katrin Wehrheim and Dietmar Salamon are working on a program to prove this assumption. ${\ displaystyle \ Sigma}$ ${\ displaystyle \ Sigma}$ ${\ displaystyle \ Sigma}$

Links to mirror symmetry

The homological mirror symmetry conjecture (homological mirror symmetry) by Maxim Konzewitsch states the equivalence of the Lagrange FH of Lagrange submanifolds in Calabi-Yau manifolds and the Ext groups of coherent sheaves on the Mirror-Calabi-Yau manifold ahead. The Floer chain groups are more interesting than the FH groups. Similar to the “pair-of-pants product”, -Gones can be formed from strings of pseudoholomorphic curves. These structures satisfy the -relations and thus make the category of all Lagrange submanifolds (without obstructions) in a symplectic manifold into a -category, called the Fukaya category . ${\ displaystyle X}$ ${\ displaystyle n}$ ${\ displaystyle A _ {\ infty}}$ ${\ displaystyle A _ {\ infty}}$

More precisely, additional structures have to be added to the Lagrangian manifold, namely a gradation and a spin structure (analogous to physics called “brane”). Then the conjecture says that there is a derived Morita equivalence between the Fukaya category of the Calabi-Yau spaces and the dg category of the derived category of the 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 cobordisms between them. It originally comes from Jakow Eliaschberg , Alexander Givental and Helmut Hofer . Like its sub-complexes, the rational symplectic field theory and the contact homology, it is defined as the homology of differential algebras that are generated by closed paths of Reeb vector fields of a contact form . The “differential” counts here certain pseudo-holomorphic curves in the cylinder over the contact manifold , the trivial examples of which are the branched overlays 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 whose differentials only count pseudo-holomorphic cylinders. However, due to the presence of pseudo-holomorphic disks, the cylindrical contact homology is not always defined. If it is defined, it can be seen as a (slightly modified) “Morse homology” of the action functional on the loop space, which assigns the integral of a contact shape over this loop to a loop. “Reeb tracks” are the critical points of this functional. ${\ displaystyle \ mathbb {R} \ times M}$ ${\ displaystyle M}$

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

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

Similarly, an analogue to “embedded contact homology” (ECH) can be defined for the mapping tori of symplectomorphisms of a surface (also with a border), the “periodic FH”, which generalizes the SFH of surface symplectomorphisms. It is presumably linked to the ECH.

Floer homotopy

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

Advancement of techniques

Many of these FHs have not been completely and strictly constructed, and many presumed equivalences are still open. Problems arise from technical difficulties such. B. in the compactification of the modular spaces of the pseudo-alcoholomorphic curves. Together with Kris Wysocki and Eduard Zehnder, Hofer developed new techniques with their theories of polyfolds and the “generalized Fredholm theory”.

calculation

Floer homologies (FH) are generally difficult to calculate 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 different classes of 3-manifolds, and its connection with other invariants has been examined.

literature

Books and reviews:

Michael Atiyah : New invariants of 3- and 4-dimensional manifolds. In: The mathematical heritage of Hermann Weyl. (Proceedings of the Symposium on the Mathematical Heritage of Hermann Weyl. Held at the Duke University, Durham, North Carolina, May 12-16, 1987) (= Proceedings of Symposia in pure Mathematics. Vol. 48). American Mathematical Society, Durham NC 1988, ISBN 0-8218-1482-6 , pp. 285-299.
Augustin Banyaga, David Hurtubise Lectures on Morse Homology (= Kluwer Texts in the Mathematical Sciences. Vol. 29). Kluwer Academic Publishers, Dordrecht et al. 2004, ISBN 1-4020-2695-1 .
Simon K. Donaldson , M. Furuta, D. Kotschick Floer Homology groups in Yang-Mills theory (= Cambridge Tracts in Mathematics. Vol. 147). Cambridge University Press, Cambridge et al. 2002, ISBN 0-521-80803-0 .
David A. Ellwood, Peter S. Ozsvath , Andras I. Stipsicz, Zoltan Szabo (Eds.): Floer Homology, Gauge Theory, and Low-dimensional. Proceedings of the Clay Mathematics Institute 2004 Summer School, Alfréd Rényi Institute of Mathematics, Budapest, Hungary, June 5-26, 2004 (= Clay Mathematics Proceedings. Vol. 5). American Mathematical Society et al., Providence RI 2006, ISBN 0-8218-3845-8 .
Dusa McDuff , Dietmar Salamon : Introduction to Symplectic Topology (= Oxford Mathematical Monographs ). 2nd edition. Clarendon Press, Oxford 1998, ISBN 0-19-850451-9 .
Dusa McDuff: Floer theory and low dimensional topology. In: Bulletin of the American Mathematical Society. Vol. 43, No. 1, 2006, ISSN 0273-0979 , 25-42, (PDF; 323 kB) .
Matthias Schwarz: Morse Homology (= Progress in Mathematics. Vol. 111). Birkhäuser, Basel et al. 1993, ISBN 3-7643-2904-1 .

Items:

Andreas Floer: The irregularized gradient flow of the symplectic action. In: Communications on Pure and Applied Mathematics. Vol. 41, No. 6, 1988, ISSN 0010-3640 , pp. 775-813, doi : 10.1002 / cpa . 3160410603 .

Andreas Floer: An instanton-invariant for 3-manifolds. In: Communications in Mathematical Physics. Vol. 118, No. 2, 1988, ISSN 0010-3616 , pp. 215-240, Project Euclid .

Andreas Floer: Morse theory for Lagrangian intersections. In: Journal of Differential Geometry. Vol. 28, 1988, ISSN 0022-040X , pp. 513-547, online (PDF; 357 kB) .

Andreas Floer: Cuplength estimates on Lagrangian intersections. In: Communications on Pure and Applied Mathematics. Vol. 42, No. 4, 1989, pp. 335-356, doi : 10.1002 / cpa . 3160420402 .

Andreas Floer: Symplectic fixed points and holomorphic spheres. In: Communications in Mathematical Physics. Vol. 120, No. 4, 1989, pp. 575-611, doi : 10.1007 / BF01260388 .

Andreas Floer: Witten's complex and infinite dimensional Morse Theory. In: Journal of Differential Geometry. Vol. 30, 1989, pp. 202-221.

Mikhail Gromov : Pseudo holomorphic curves in symplectic manifolds. In: Inventiones Mathematicae . Vol. 82, 1985, pp. 307-347, online (PDF; 2.4 MB) .

Helmut Hofer , Kris Wysocki, Eduard Zehnder : A General Fredholm Theory I: A Splicing-Based Differential Geometry. online .

Peter Ozsváth, Zoltán Szabó: On the Heegaard Floer homology of branched double-covers. In: Advances in Mathematics. Vol. 194, No. 1, 2005, pp. 1-33, doi : 10.1016 / j.aim.2004.05.008 , a preprint .

Web links

Individual evidence

↑ According to Ozsvath-Szabo (2005), a variant of the Khovanov homology is linked via a spectral sequence with the Heegaard-Floer homology of an overlay branched along a node .
↑ Peter Kronheimer, Tomasz Mrowka: Monopoles and Three-Manifolds (= New Mathematical Monographs. Vol. 10). Cambridge University Press, Cambridge 2007, ISBN 978-0-511-54311-1 , review in the Zentralblatt .

[1] According to Ozsvath-Szabo (2005), a variant of the Khovanov homology is linked via a spectral sequence with the Heegaard-Floer homology of an overlay branched along a node .

[2] Peter Kronheimer, Tomasz Mrowka: Monopoles and Three-Manifolds (= New Mathematical Monographs. Vol. 10). Cambridge University Press, Cambridge 2007, ISBN 978-0-511-54311-1 , review in the Zentralblatt .