Floer homology for almost Hamiltonian isotopies

Banyaga, Augustin; Saunders, Christopher

doi:10.1016/j.crma.2006.01.001

Differential Geometry

Floer homology for almost Hamiltonian isotopies
[Homologie de Floer pour les isotopies presques hamiltonniennes]

Présenté par : Charles-Michel Marle

Banyaga, Augustin ¹ ; Saunders, Christopher ²

¹ Department of Mathematics, Pennsylvania State University, 218 McAllister Building, University Park, PA 16803, USA
² Department of Mathematics, Westminster College, Fulton, MO 65251, USA

Comptes Rendus. Mathématique, Tome 342 (2006) no. 6, pp. 417-420.

Résumé
Abstract

Seidel a introduit un homomorphisme du groupe fondamental $π_{1} (Ham (M))$ du groupe des difféomorphismes Hamiltoniennes de certaines variétés symplectiques compactes $(M, ω)$ dans un quotient du groupe $Aut ({HF}_{*} (M, ω))$ des automorphismes de l'homologie de Floer ${HF}_{*} (M, ω)$ . Nous démontrons que si deux lacets Hamiltoniennes representent le même élément dans $π_{1} (Diff (M))$ , alors les images par l'homomorphisme de Seidel de leurs classes dans $π_{1} (Ham (M))$ coïncident (un phénomène de rigidité). La preuve consiste à montrer que l'homologie de Floer peut être définie en utilisant des isotopies presques Hamiltoniennes, c'est-à-dire des isotopies qui sont homotopes, relativement aux extrémités à des isotopies Hamiltoniennes.

Seidel introduced a homomorphism from the fundamental group $π_{1} (Ham (M))$ of the group of Hamiltonian diffeomorphisms of certain compact symplectic manifolds $(M, ω)$ to a quotient of the automorphism group $Aut ({HF}_{*} (M, ω))$ of the Floer homology ${HF}_{*} (M, ω)$ . We prove a rigidity property: if two Hamiltonian loops represent the same element in $π_{1} (Diff (M))$ , then the image under the Seidel homomorphism of their classes in $π_{1} (Ham (M))$ coincide. The proof consists in showing that Floer homology can be defined by using ‘almost Hamiltonian’ isotopies, i.e. isotopies that are homotopic relatively to endpoints to Hamiltonian isotopies.

Reçu le : 2005-09-26
Accepté le : 2006-01-04
Publié le : 2006-02-07

DOI : 10.1016/j.crma.2006.01.001

Affiliations des auteurs :

Banyaga, Augustin ¹ ; Saunders, Christopher ²

¹ Department of Mathematics, Pennsylvania State University, 218 McAllister Building, University Park, PA 16803, USA
² Department of Mathematics, Westminster College, Fulton, MO 65251, USA

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     title = {Floer homology for almost {Hamiltonian} isotopies},
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     pages = {417--420},
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Banyaga, Augustin; Saunders, Christopher. Floer homology for almost Hamiltonian isotopies. Comptes Rendus. Mathématique, Tome 342 (2006) no. 6, pp. 417-420. doi : 10.1016/j.crma.2006.01.001. http://www.numdam.org/articles/10.1016/j.crma.2006.01.001/

Bibliographie
Cité par

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[5] Salamon, D. Lectures on Floer homology, Symplectic Geometry and Topology, Park City, UT, 1997, IAS/Park City Math. Ser., vol. 7, Amer. Math. Soc., Providence, RI, 1999, pp. 143-229

[6] Seidel, P. $π_{1}$ of symplectic automorphism groups and invertibles in quantum homology rings, Geom. Funct. Anal., Volume 7 (1997), pp. 1046-1095

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