Lalonde, François; McDuff, Dusa; Polterovich, Leonid Topological rigidity of Hamiltonian loops and quantum homology. (English) Zbl 0907.58004 Invent. Math. 135, No. 2, 369-385 (1999). This paper studies the question of when a loop \(\phi= \{\phi_t\}_{0\leq t\leq 1}\) in the group \(\text{Symp} (M,\omega)\) of symplectomorphisms of a symplectic manifold \((M,\omega)\) 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 \(\phi\) is Hamiltonian with respect to \(\omega\), and if \(\phi'\) is a small perturbation of \(\phi\) that preserves another symplectic form \(\omega'\), then \(\phi'\) is Hamiltonian with respect to \(\omega'\). This allows us to get some new information on the structure of the flux group, i.e., the image of \(\pi_1(\text{Symp} (M,\omega))\) 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 \(M\). Reviewer: Leonid Polterovich (Zürich) Cited in 2 ReviewsCited in 24 Documents MSC: 58D05 Groups of diffeomorphisms and homeomorphisms as manifolds 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) Keywords:topological rigidity; Hamiltonian loops; symplectomorphisms; symplectic manifold; quantum homology PDFBibTeX XML Full Text: DOI arXiv