Date of Award:
5-1968
Document Type:
Thesis
Degree Name:
Master of Science (MS)
Department:
Mathematics and Statistics
Department name when degree awarded
Mathematics
Committee Chair(s)
Lawrence O. Cannon
Committee
Lawrence O. Cannon
Abstract
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 M3 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 φℓk: Gk → Gℓ, 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.
Checksum
59c702a1b7f1c7b97622dbcd6159daab
Recommended Citation
Riebe, Norman William, "The Fundamental Groups of the Complements of Some Solid Horned Spheres" (1968). All Graduate Theses and Dissertations, Spring 1920 to Summer 2023. 6821.
https://digitalcommons.usu.edu/etd/6821
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