The space of mechanical motions of a system has the structure of an infinite-dimensional group. When the system is described by a symplectic manifold, the mechanical motions correspond to Hamiltonian symplectic diffeomorphisms. Hofer in the 1990s defined a remarkable metric on this group, which in a sense measures the minimal energy needed to generate a given mechanical motion. The resulting geometry has been successfully studied using Gromov's theory of pseudo-holomorphic curves in the symplectic manifold. In this thesis we further extend the relationship between the theory pseudo-holomorphic curves (Gromov-Witten theory) and Hofer geometry. We define natural characteristic classes on the loop space of the Hamiltonian diffeomorphism group, with values in the quantum homology of the manifold. In particular, we show that there is a natural graded ring homomorphism from the Pontryagin ring of the homology of the loop space to the quantum homology of the symplectic manifold. These classes can be viewed as giving a kind of virtual Morse theory for the Hofer length functional on the loop space and give rise to some difficult and interesting questions. We compute these classes, in some cases, by Morse-Bott type of techniques and give applications to topology and Hofer geometry of the group of Hamiltonian diffeomorphisms.
|School:||State University of New York at Stony Brook|
|School Location:||United States -- New York|
|Source:||DAI-B 71/05, Dissertation Abstracts International|
|Keywords:||Gromov-Witten theory, Hamiltonian diffeomorphism, Pseudoholomorphic curves, Quantum characteristic classes|
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