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The Poincaré Conjecture

The central theme for this paper is provided by the following three statements:
(1) Every compact connected 1-manifold is S1.
(2) Every compact connected simply connected 2-manifold is S2.
(3) Every compact connected simply connected 3-manifold is S3.
We provide proofs of statements (1) and (2). The veracity of the third statement, the Poincaré Conjecture, has not been determined. It is known that should a counter-example exist it can be found by removing from S3 a finite collect ion of solid tori and sewing them back differently. We show that it is not possible to find a counterexample by removing from S3 a single solid torus of twist knot type or torus knot type and sewing it back differently. We treat as special cases a solid torus of trivial knot type and trefoil knot type.

Identiferoai:union.ndltd.org:UTAHS/oai:digitalcommons.usu.edu:etd-7956
Date01 May 1971
CreatorsPeck, Joseph D.
PublisherDigitalCommons@USU
Source SetsUtah State University
Detected LanguageEnglish
Typetext
Formatapplication/pdf
SourceAll Graduate Theses and Dissertations
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