Floer homology for almost Hamiltonian isotopies

Abstract

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. To cite this article: A. Banyaga, C. Saunders, C. R. Acad. Sci. Paris, Ser. I 342 (2006).