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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 Urbana-Champaign
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 well-known 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 2-torus 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:2012-02-06
Genre:thesis
URI:http://hdl.handle.net/2142/29636
Rights Information:Copyright 2011 Daniel Morton
Date Available in IDEALS:2012-02-06
Date Deposited:2011-12


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