Differential Geometry
Floer homology for almost Hamiltonian isotopies

Presented by: Charles-Michel Marle

Banyaga, Augustin1; Saunders, Christopher2
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.

Seidel introduced a homomorphism from the fundamental group π1(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 π1(Diff(M)), then the image under the Seidel homomorphism of their classes in π1(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 π1(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 π1(Diff(M)), alors les images par l'homomorphisme de Seidel de leurs classes dans π1(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:
Accepted:
Published online:
DOI: 10.1016/j.crma.2006.01.001
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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Banyaga, Augustin; Saunders, Christopher. 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://www.numdam.org/articles/10.1016/j.crma.2006.01.001/

[1] Banyaga, A. The Structure of Classical Diffeomorphism Groups, Kluwer Academic Press, 1997

[2] Floer, A. Symplectic fixed points and holomorphic spheres, Comm. Math. Phys., Volume 120 (1989), pp. 575-611

[3] Hofer, H.; Salamon, D. Floer homology and Novikov rings, The Floer Memorial Volume, Birkhäuser, Basel, 1995, pp. 483-524

[4] Lalonde, F.; McDuff, D.; Polterovich, L. Topological rigidity of Hamiltonian loops and quantum homology, Invent. Math., Volume 135 (1999) no. 2, pp. 369-385

[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. π1 of symplectic automorphism groups and invertibles in quantum homology rings, Geom. Funct. Anal., Volume 7 (1997), pp. 1046-1095

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