Applications of the h-Cobordism Theorem

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Authors
Burkemper, Matthew Bryan
Issue Date
2014-05
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Thesis
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en_US
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Abstract

We provide an exposition of J. Milnor's proof of the h-Cobordism Theorem. This theorem states that a smooth, compact, simply connected n-dimensional manifold W with n greater than or equal to 6, whose boundary boundaryW consists of a pair of closed simply connected (n-1)-dimensional manifolds M0 and M1 and whose relative integral homology groups H(W,M0) are all trivial, is diffeomorphic to the cylinder M0 x [0, 1]. The proof makes heavy use of Morse Theory and in particular the cancellation of certain pairs of Morse critical points of a smooth function. We pay special attention to this cancellation and provide some explicit examples. An important application of this theorem concerns the generalized Poincare conjecture, which states that a closed simply connected n-dimensional manifold with the integral homology of the n-dimensional sphere is homeomorphic to the sphere. We discuss the proof of this conjecture in dimension n greater than or equal to 6, which is a consequence of the h-Cobordism Theorem.

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Thesis (M.S.)--Wichita State University, Fairmount College of Liberal Arts and Sciences, Dept. of Mathematics, Statistics, and Physics
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Wichita State University
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Copyright 2014 Matthew Bryan Burkemper
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