Floer Homologies, with Applications

Abstract

Floer invented his theory in the mid eighties in order to prove the Arnol'd conjectures on the number of fixed point 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. Other Floer homologies are of topological nature. Among their applications are Property P for knots and the construction of compact topological manifolds of dimension greater than five that are not triangulisable. 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. 14 4.

The Beginnings of Symplectic Topology in Bochum in the Early Eighties

“…The relevant broken connecting orbits at the ends therefore appear in pairs and so ∂ • ∂ = 0 mod 2. The homology of the complex (C, ∂), Andreas Floer's methods, ideas, and constructions were a crucial break-through and continue to influence symplectic topology and Hamiltonian dynamics enormously, see for instance the subsequent survey [1].…”

“…Inspired by Conley's continuation theorem for the Conley index, Floer showed in a second step that his Floer homology is independent of the choice of H and J. He takes two Hamiltonians (H α , J α , x α ) and (H β , J β , x β ) with the associated almost complex structures and generators of the corresponding chain complexes, and defines a clever homotopy between the Hamiltonians and almost complex structures, which satisfies, in particular, Andreas Floer's methods, ideas, and constructions were a crucial break-through and continue to influence symplectic topology and Hamiltonian dynamics enormously, see for instance the subsequent survey [1].…”

Three Examples in the Dynamical Systems Theory

“…Yakovenko that while "mathematical anecdotes mention great mathematicians whom examples only distracted from developing general theories", "Arnold was the opposite: examples were the alpha and omega of his approach" [36]. 1 Another outstanding mathematician of the last century, I.R. Shafarevich, wrote that "the meaning of a mathematical notion is by no means confined to its formal definition; in fact, it may be rather better expressed by a (generally fairly small) sample of the basic examples, which serve the mathematician as the motivation and the substantive definition, and at the same time as the real meaning of the notion" [30].…”

A compactness result for non-local unregularized gradient flow lines

“…Although we nowadays have many examples of Floer homologies, see for instance [2], it is still a difficult issue to say precisely what an unregularized gradient flow equation is. New insight into this question comes from the recent discovery by Hofer, Wysocki, and Zehnder [15] of new smooth structures in infinite dimensions.…”