Date of Award:
Master of Science (MS)
Mathematics and Statistics
Lawrence O. Cannon
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)
Riebe, Norman William, "The Fundamental Groups of the Complements of Some Solid Horned Spheres" (1968). All Graduate Theses and Dissertations. 6821.