One of the methods used for the construction of the classical Alexander horned sphere leads naturally to generalization to horned spheres of higher order. Let M2, denote the Alexander horned sphere. This is a 2-horned sphere of order 2. Denote by M 3 and M4, two 2-horned spheres of orders 3 and 4, respectively, constructed by such a generalization.
The fundamental groups of the complements of M2, M3, and M4 are derived, and representations of these groups onto the Alternating Group, A5, are found. The form of the presentations of these fundamental groups leads to a more general class of groups, denoted by Gk, k ≥ 2. A set of homomorphisms ϴkl : Gk, k ≥ l ≥ 2 is found, which has a clear geometric meaning as applied to the groups G2, G3, and G4.
Two theorems relating to direct systems of non-abelian groups are proved and applied to the groups Gk. The implication of these theorems is that the groups Gk, k≥2 are all free groups of countably infinite rank and that the embeddings of M2, M3, and M4 in E3 cannot be distinguished by means of fundamental groups. *33 pages)
| Identifer | oai:union.ndltd.org:UTAHS/oai:digitalcommons.usu.edu:etd-7892 |
| Date | 01 May 1968 |
| Creators | Riebe, Norman William |
| 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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