TY - JOUR

T1 - Topological properties of Hamiltonian circle actions

AU - McDuff, Dusa

AU - Tolman, Susan

N1 - Copyright:
Copyright 2006 Elsevier B.V., All rights reserved.

PY - 2006

Y1 - 2006

N2 - This paper studies Hamiltonian circle actions, that is, circle subgroups of the group Ham(M, ω) of Hamiltonian symplectomorphisms of a closed symplectic manifold (M, ω). Our main tool is the Seidel representation of n1(Ham(M, ω)) in the units of the quantum homology ring. We show that if the weights of the action at the points at which the moment map is a maximum are sufficiently small, then the circle represents a nonzero element of n1(Ham(M, ω)). Further, if the isotropyhas order at most two and the circle contracts in Ham(M, ω), then various symmetry properties hold. For example, the image of the normalized moment map is a symmetric interval [-a, a]. If the action is semifree (i.e., the isotropy weights are 0 or ±1), then we calculate the leading order term in the Seidel representation, an important technical tool in understanding the quantum cohomology of manifolds that admit semifree Hamiltonian circle actions. If the manifold is toric, we use our results about this representation to describe the basic multiplicative structure of the quantum cohomology ring of an arbitrary toric manifold. There are two important technical ingredients; one relates the equivariant cohomology of M to the Morse flow of the moment map, and the other is a version of the localization principle for calculating Gromov-Witten in variants on symplectic manifolds with S1-actions.

AB - This paper studies Hamiltonian circle actions, that is, circle subgroups of the group Ham(M, ω) of Hamiltonian symplectomorphisms of a closed symplectic manifold (M, ω). Our main tool is the Seidel representation of n1(Ham(M, ω)) in the units of the quantum homology ring. We show that if the weights of the action at the points at which the moment map is a maximum are sufficiently small, then the circle represents a nonzero element of n1(Ham(M, ω)). Further, if the isotropyhas order at most two and the circle contracts in Ham(M, ω), then various symmetry properties hold. For example, the image of the normalized moment map is a symmetric interval [-a, a]. If the action is semifree (i.e., the isotropy weights are 0 or ±1), then we calculate the leading order term in the Seidel representation, an important technical tool in understanding the quantum cohomology of manifolds that admit semifree Hamiltonian circle actions. If the manifold is toric, we use our results about this representation to describe the basic multiplicative structure of the quantum cohomology ring of an arbitrary toric manifold. There are two important technical ingredients; one relates the equivariant cohomology of M to the Morse flow of the moment map, and the other is a version of the localization principle for calculating Gromov-Witten in variants on symplectic manifolds with S1-actions.

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U2 - 10.1155/IMRP/2006/72826

DO - 10.1155/IMRP/2006/72826

M3 - Article

AN - SCOPUS:33645655860

VL - 2006

JO - International Mathematics Research Papers

JF - International Mathematics Research Papers

SN - 1687-3017

M1 - 72826

ER -