×

zbMATH — the first resource for mathematics

Cuplength estimates on Lagrangian intersections. (English) Zbl 0683.58017
This article presents a proof of the following version of Arnold’s conjecture: “Let L be a compact Lagrangian submanifold of a symplectic manifold P, such that \(\pi_ 2(P,L)=0\). Then for any exact diffeomorphism \(\phi\), the number of intersections of \(\phi\) (L) with L is greater than or equal to the \(Z_ 2\)-cuplength of L.” This theorem implies the following corollary: “If P is a compact symplectic manifold with \(\pi_ 2(P)=0\), then the number of fixed points of any exact diffeomorphism of P is greater than or equal to the \(Z_ 2\)-cuplength of P.”
Reviewer: V.Perlick

MSC:
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Arnold, Mathematical Methods of Classical Mechanics (1978) · Zbl 0386.70001 · doi:10.1007/978-1-4757-1693-1
[2] Arnold, Sur une proprietĂ© topologique des applications globalement canoniques de la mecanique classique, C.R. Acad. Sci. Paris 261 pp 3719– (1965)
[3] Conley, Isolated invariant sets and the Morse index 38 (1978) · Zbl 0397.34056 · doi:10.1090/cbms/038
[4] Conley, Morse type index theory for flows and periodic solutions for Hamiltonian equations, Comm. Pure Appl. Math. 34 pp 207– (1984) · Zbl 0559.58019
[5] Floer, A refinement of the Conley index and an application to the stability of hyperbolic invariant sets, Erg. Theory and Dyn. Syst. 7 pp 93– (1987) · Zbl 0633.58040 · doi:10.1017/S0143385700003825
[6] Floer, Morse theory for Lagrangian intersections, J. Diff. Geom. 28, No. 3 (1988) · Zbl 0674.57027
[7] Floer, The unregularized gradient flow of the symplectic action, Comm. Pure Appl. Math. 41 pp 775– (1988) · Zbl 0633.53058
[8] Floer, A relative Morse index for the symplectic action, Comm. Pure Appl. Math. 41 pp 393– (1988) · Zbl 0633.58009
[9] Floer, Witten’s complex for arbitrary coefficients and an application to Lagrangian intersections, J. Diff. Geom. 29 (1989)
[10] Gromov, Pseudo-holomorphic curves in symplectic manifolds, Inv. Math. 82 pp 307– (1985) · Zbl 0592.53025
[11] Hofer, Lagrangian embeddings and critical point theory, Ann. Inst. H. PoincarĂ©, Analyse Nonlineaire 2 pp 407– (1985)
[12] Hofer, H., On Ljusternik-Schnirelman theory for Lagrangian intersections, preprint, Rutgers University. · Zbl 0669.58006
[13] Laudenbach, Persistance d’intersection avec la section nulle en cours d’une isotopie hamiltonienne dans un fibre cotangent, Inv. Math. 82 pp 349– (1985) · Zbl 0592.58023
[14] Milnor, Ann. of Math. Studies 76 (1974)
[15] Spanier, Algebraic Topology (1966)
[16] Weinstein, Symplectic manifolds and their Lagrangian submanifolds, Adv. in Math. 6 pp 329– (1971) · Zbl 0213.48203
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.