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Zbl 0621.58018
Sikorav, Jean-Claude
Un problème de disjonction par isotopie symplectique dans un fibré cotangent. (A problem of disjunction by symplectic isotopy in a cotangent bundle). (French)
[J] Ann. Sci. Éc. Norm. Supér. (4) 19, No. 4, 543-552 (1986). ISSN 0012-9593

The main problem which the author wants to solve is that of finding lower bounds for the number of intersection points of M with $\phi\sb 1(M)$, where $\phi\sb t: M\to T\sp*M$ (0$\le t\le 1$, $\phi\sb 0=id)$ is a symplectic isotopy. His main theorem proves the existence of a bijection between the points of $M\cap \phi\sb 1(M)$ and the zeroes of a closed 1- form on some product $M\times V$, such that the transversal intersections correspond to Morse type zeros. Then, lower bounds of the number of Morse zeros of a closed 1-form due to {\it S. P. Novikov} [Sov. Math., Dokl. 24, 222-226 (1981); translation from Dokl. Akad. Nauk SSSR 260, 31-35 (1981; Zbl 0505.58011)] are studied, in conjunction with {\it M. Sh. Farber}'s results [Funct. Anal. Appl. 19, 40-48 (1985); translation from Funkts. Anal. Prilozh. 19, No.1, 49-59 (1985; Zbl 0603.58030)]. This allows the author to give lower bounds of the desired type under some supplementary hypotheses: $\phi\sb 1(M)$ is transversal to M, dim $M\ge 6$, $\pi\sb 1(M)={\bbfZ}$.
[I.Vaisman]

MSC 2000:: *37J99 Finite-dimensional Hamiltonian etc. systems
57R52 Isotopy (differential topology)

Keywords: intersection points; zeroes of a closed 1-form

Citations: Zbl 0505.58011; Zbl 0603.58030

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