×

zbMATH — the first resource for mathematics

Symplectic fixed points, the Calabi invariant and Novikov homology. [Appendix C in collaboration with Lê Tu Quôc Thang]. (English) Zbl 0822.58019
Let \((M,\omega)\) be a compact symplectic manifold. Then we can associate each path joining the identity element and an element \(g_ 1\) in a symplectomorphism group of \((M,\omega)\) with a value in \(H^ 1 (M,R)\) which is called the Calabi invariant of this element \(g_ 1\) (Calabi has shown that this value depends only on the relative homotopy class of path). A symplectomorphism \(g_ 1\) is called homologous to zero (and later on, the terminology “exact” has been widely accepted) if it can be generated by a time-dependent Hamiltonian flow on \(M\). Banyaga has shown that a symplectomorphism is homologous to zero if and only if its Calabi invariant is zero. Thus the famous Arnold conjecture on the symplectic fixed points of an exact symplectomorphism is related to the class of symplectomorphisms with zero Calabi invariant. In this note the authors prove an analog of the Arnold conjecture for any symplectomorphism in the identity component of the symplectomorphism group on a monotone symplectic manifold. Thus the Morse type inequality is replaced by Morse-Novikov inequality related to the Calabi invariant of a considered symplectomorphism. They used the Floer homology but on non-compact covering of \(M\), thus the usually needed energy estimate in Floer homology is quite non-trivial in their situation. They got this energy estimate by a special choice of homomorphism between two Floer- Novikov homology groups corresponding to different Calabi invariant. This choice and some other technical difficulty prevent them to prove the theorem on the class of weakly monotone symplectic manifolds, on which the Arnold conjecture was confirmed by the second author, and the Floer- Novikov homology also was defined in the paper.
This note also consists of three appendices. In the first one they prove that two embedded loops in \((M,\omega)\) are equivalent under the action of symplectomorphism group if and only if their Poincaré invariant are the same. The second provides a proof of a transversality result needed to construct Floer homology for the time-independent Hamiltonian. The last one in collaboration with Lê Tu Quôc Thang is a short exposition in Novikov homology theory.

MSC:
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
PDF BibTeX XML Cite
Full Text: DOI