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Title:  GKM manifolds with low Betti numbers 
Author(s):  Morton, Daniel 
Director of Research:  Tolman, Susan 
Doctoral Committee Member(s):  Kerman, Ely; Tolman, Susan; Schenck, Henry K. 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  GKM Manifolds
GKM Graphs Symplectic Geometry Symplectic Manifolds Torus Actions 
Abstract:  A GKM manifold is a symplectic manifold with a torus action such that the fixed points are isolated and the isotropy weights at the fixed points are linearly independent. Each GKM manifold has a GKM graph which contains much of the topological information of the manifold, in particular the equivariant cohomology and Chern classes. We are interested in the case where the torus action is Hamiltonian. In this thesis we will consider the case where the GKM graphs are complete. When the dimension of the torus action is sufficiently large, we can completely classify the complete, in the graph theoretic sense, GKM graphs, and thus completely describe the cohomology rings and Chern classes of the associated ”minimal” GKM manifolds. For each possible cohomology ring and total Chern class we can find a wellknown GKM manifold having that ring and class. If we put some restrictions on the allowable subgraph, and thus restrict the allowable submanifolds, then we can completely classify the possible cohomology rings and Chern classes of minimal GKM manifolds. We will also consider one of the cases where the GKM graph is not complete. In the case of six dimensional symplectic manifolds whose GKM structure comes from a Hamiltonian 2torus action we can also completely classify all the possible GKM graphs, and thus all the possible cohomology rings and Chern classes. Once again, for each possible cohomology ring and total Chern class, we can construct a manifold having that ring and class. 
Issue Date:  20120206 
Genre:  Dissertation / Thesis 
URI:  http://hdl.handle.net/2142/29636 
Rights Information:  Copyright 2011 Daniel Morton 
Date Available in IDEALS:  20120206 
Date Deposited:  201112 
This item appears in the following Collection(s)

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois 
Dissertations and Theses  Mathematics