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.
| Identifer | oai:union.ndltd.org:UTAHS/oai:digitalcommons.usu.edu:etd-7956 |
| Date | 01 May 1971 |
| Creators | Peck, Joseph D. |
| Publisher | DigitalCommons@USU |
| Source Sets | Utah State University |
| Detected Language | English |
| Type | text |
| Format | application/pdf |
| Source | All Graduate Theses and Dissertations |
| Rights | Copyright for this work is held by the author. Transmission or reproduction of materials protected by copyright beyond that allowed by fair use requires the written permission of the copyright owners. Works not in the public domain cannot be commercially exploited without permission of the copyright owner. Responsibility for any use rests exclusively with the user. For more information contact digitalcommons@usu.edu. |
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