Floer homology for almost Hamiltonian isotopies

Augustin Banyaga; Christopher Saunders

doi:10.1016/j.crma.2006.01.001

Differential Geometry

Floer homology for almost Hamiltonian isotopies

Presented by: Charles-Michel Marle

Augustin Banyaga ¹; Christopher Saunders 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, Volume 342 (2006) no. 6, pp. 417-420.

Abstract
Résumé

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.

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.

Received: 2005-09-26
Accepted: 2006-01-04
Published online: 2006-02-07

DOI: 10.1016/j.crma.2006.01.001

Author's affiliations:

Augustin Banyaga ¹; Christopher Saunders 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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     author = {Augustin Banyaga and Christopher Saunders},
     title = {Floer homology for almost {Hamiltonian} isotopies},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {417--420},
     publisher = {Elsevier},
     volume = {342},
     number = {6},
     year = {2006},
     doi = {10.1016/j.crma.2006.01.001},
     language = {en},
}

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Augustin Banyaga; Christopher Saunders. Floer homology for almost Hamiltonian isotopies. Comptes Rendus. Mathématique, Volume 342 (2006) no. 6, pp. 417-420. doi : 10.1016/j.crma.2006.01.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.01.001/

References
Cited by

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