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Floer homologies, with applications. (English) Zbl 1433.53115

Authors’ abstract: Floer invented his theory in the mid eighties in order to prove the Arnol’d conjectures on the number of fixed points of Hamiltonian diffeomorphisms and Lagrangian intersections. Over the last thirty years, many versions of Floer homology have been constructed. In symplectic and contact dynamics and geometry they have become a principal tool, with applications that go far beyond the Arnol’d conjectures: The proof of the Conley conjecture and of many instances of the Weinstein conjecture, rigidity results on Lagrangian submanifolds and on the group of symplectomorphisms, lower bounds for the topological entropy of Reeb flows and obstructions to symplectic embeddings are just some of the applications of Floer’s seminal ideas. This is by no means a comprehensive survey on the presently known Floer homologies and their applications. Such a survey would take several hundred pages. We just describe some of the most classical versions and applications, together with the results that we know or like best. The text is written for non-specialists, and the focus is on ideas rather than generality. Two intermediate sections recall basic notions and concepts from symplectic dynamics and geometry. Floer built Floer homologies in two situations: For the action functional of classical mechanics on the space of curves on a symplectic manifold and for the Chern-Simons functional on the space of \(\mathrm{SU}(2)\) connections on a 3-dimensional oriented integral homology sphere \(Y\). The first theory is called Hamiltonian or Lagrangian Floer homology-depending on whether one considers closed or open curves- and proves at once important cases of Arnol’d’s conjectures on the number of fixed points of Hamiltonian diffeomorphisms and of Lagrangian intersection points. The second one, called instanton homology, refines a topological invariant: Its Euler characteristic is twice the Casson invariant of \(Y\). By now, many other Floer homologies have been constructed, both of symplectic and of topological nature, and they have many more applications to dynamics and geometry.
Our survey starts with a discussion of Morse homology on a finite dimensional compact manifold, since this homology is the model for every Floer homology. All features of Morse homology have a counterpart in every Floer homology, and the first thing one should do if one wants to check if a Floer homology has a certain property is to see if this property holds for Morse homology. In Sect. 4 we then describe in some detail the arguably easiest version of Floer homology, namely Hamiltonian Floer homology for symplectically aspherical closed symplectic manifolds. Many aspects and technical issues of other Floer homologies are analogous, so that in our subsequent discussion of other Floer homologies we can give less details on the construction and focus on their applications.
Symplectic homology (Section 6) is an invariant of suitable subsets of a symplectic manifold. It leads to a proof of the Weinstein conjecture on the existence of a closed Hamiltonian orbit on an energy surface, and it provides obstructions to the existence of Lagrangian submanifolds and of certain symplectic embeddings. Lagrangian Floer homology (Section 7) gives lower bounds on the number of intersections of certain pairs of Lagrangian submanifolds and can be used to prove that for many contact manifolds all Reeb flows must have positive topological entropy.
Finally, in Section 8 we describe two Floer homologies for contact manifolds. Contact homology can be used to distinguish contact structures on compact contact manifolds. Embedded contact homology, that is defined for three-dimensional compact contact manifolds, can be used to prove the Weinstein conjecture in dimension three, and it gives rise to numerical invariants that yield very fine symplectic embedding obstructions in dimension four and imply the \(C^{\infty}\) closing lemma for Hamiltonian diffeomorphisms on surfaces. These are all “symplectic” Floer homologies and applications to symplectic and contact dynamics and geometry.
Other Floer homologies are of topological nature. They have applications to the structure of manifolds, like Property P for knots, the question which compact 3-manifolds can be obtained by Dehn surgery of \(S^3\) on which knot, and the existence of topological manifolds of dimension 5 that admit no triangulation.

MSC:

53D40 Symplectic aspects of Floer homology and cohomology
53D35 Global theory of symplectic and contact manifolds
53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
37J46 Periodic, homoclinic and heteroclinic orbits of finite-dimensional Hamiltonian systems

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