GKM manifolds with low Betti numbers

Morton, Daniel

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https://hdl.handle.net/2142/29636 Copy

Description

Title

GKM manifolds with low Betti numbers

Author(s)

Morton, Daniel

Issue Date

2012-02-06T20:08:06Z

Director of Research (if dissertation) or Advisor (if thesis)

Tolman, Susan

Committee Member(s)

• Kerman, Ely
• Tolman, Susan
• Schenck, Henry K.

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(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.

Graduation Semester

2011-12

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http://hdl.handle.net/2142/29636

Copyright and License Information

Copyright 2011 Daniel Morton

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