Symplectic Manifolds With Hamiltonian Circle Actions

In this thesis, we first introduce the basic definition and properties of symplectic manifolds, then we introduce symplectic circle action and Hamil-tonian circle action on a symplectic manifold. Hamiltonian circle action cor-responds to a real valued function, called moment map. Then we introduce Morse theory and prove that the moment map is a Morse function. Moreover, we discuss S1-equivariant cohomology theory, emphasize some importan-t tools and give the relation between equivariant cohomology and ordinary cohomology. Finally, we introduce equivariant Chern classes and ordinary Chern classes of a symplectic manifold.Let the circle act in a Hamiltonian fashion on a 2n-dimensional compact symplectic manifold (M, ω). We cannot always use the local information of the circle action at the fixed points to determine the global invariants of M. In this thesis, we consider the cases when the fixed point set of the circle action consists of exactly n+1 or n+2 isolated points. For the case when the circle action has exactly n+1 isolated fixed points, we obtain a basis of the integral cohomology ring of M. Our basis is consistent with the basis gotten by Tolman in another way. The basis allows us to determine the inte-gral cohomology ring and total Chern class of M from the local information of the circle action at the fixed points [10]. For the case when the circle ac-tion has exactly n+2 isolated fixed points, we use equivariant cohomology technique and Morse theory to give a basis of the integral equivariant coho-mology ring of M and their restriction to the fixed points. Then we prove that the integral cohomology ring and total Chern class of M are determined by the local information of the circle action at the fixed points. Therefore, if the S1-representations at the fixed points on M are the same as those of a stan-dard circle action on some known classical manifold, then M and this known manifold have isomorphic integral cohomology ring and total Chern class.