Topological rigidity of Hamiltonian loops and quantum homology |
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Authors: | François Lalonde Dusa McDuff Leonid Polterovich |
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Institution: | (1) Université du Québec à Montréal, Montréal, Canada (flalonde@math.uqam.ca) State University of New York at Stony Brook, Stony Brook, NY 11794-3651, USA (dusa@math.sunysb.edu) Tel-Aviv University, Tel-Aviv, Israel (polterov@math.tau.ac.il), CA |
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Abstract: | This paper studies the question of when a loop φ={φ
t
}0≤
t
≤1 in the group Symp(M,ω) of symplectomorphisms of a symplectic manifold (M,ω) is isotopic to a loop that is generated by a time-dependent Hamiltonian function. (Loops with this property are said to
be Hamiltonian.) Our main result is that Hamiltonian loops are rigid in the following sense: if φ is Hamiltonian with respect
to ω, and if φ′ is a small perturbation of φ that preserves another symplectic form ω′, then φ′ is Hamiltonian with respect
to ω′. This allows us to get some new information on the structure of the flux group, i.e. the image of π1(Symp(M,ω)) under the flux homomorphism. We give a complete proof of our result for some manifolds, and sketch the proof in general.
The argument uses methods developed by Seidel for studying properties of Hamiltonian loops via the quantum homology of M.
Oblatum 31-X-1997 & 20-III-1998 / Published online: 14 October 1998 |
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