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Title page for ETD etd-07252007-095611

Type of Document Dissertation
Author Lambert, Thomas Paul
Author's Email Address paul.lambert@vanderbilt.edu
URN etd-07252007-095611
Title On the Classification of Closed Flat Four-Manifolds
Degree PhD
Department Mathematics
Advisory Committee
Advisor Name Title
Dr. John G. Ratcliffe Committee Chair
Dr. Bruce Hughes Committee Member
Dr. Guoliang Yu Committee Member
Dr. Steven T. Tschantz Committee Member
Dr. Thomas W. Kephart Committee Member
  • flat manifolds
  • classification
  • algebraic topology
  • riemannian manifolds
  • geometry
Date of Defense 2007-05-01
Availability unrestricted
In this thesis, we discuss the classification of closed flat four-manifolds. First, we give a brief treatment of the history of this problem, with a discussion of the advantages, disadvantages, and inconsistencies among the previous classifications. Then, we discuss the process by which we reconcile the different classifications. The procedure is to exhibit presentations as isometry groups of the 74 groups corresponding to the manifolds. We determine fiber-bundle or twisted $I$-bundle decompositions of the manifolds, along with their homology and holonomy groups. Also, we determine the orientable double-covers for the nonorientable manifolds. In addition, we match our list of manifolds to those of the previous primary sources.

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