Students: Dissertation: Marcia Lynne Burdon

Marcia Lynne Burdon

ABSTRACT
Given two homotopic embeddings of a 2-polyhedron in a 3-manifold, D. Repovs asked whether the regular neighborhoods of the embeddings of the polyhedron are homeomorphic. Repovs gave an example of two homotopic embeddings of a polyhedron which have different regular neighborhoods. Repovs' question can be answered affirmatively if conditions are placed on the 2-polyhedron, the 3-manifold, or the boundaries of the regular neighborhoods of the 2-polyhedron.

Cavicchioli examined the case where the 3-manifold is a closed, prime 3-manifold which is not a fake 3-sphere. He showed that if the boundaries of the regular neighborhoods of the embeddings are homeomorphic to the 2-sphere, the regular neighborhoods of the 2-polyhedron are homeomorphic. His result doesn't require that the embeddings be homotopic.

If the assumption that the 3-manifold be prime is dropped, the connected sum of the lens spaces L(5,1) and L(5,2) gives a counter-example. It is a 3-manifold with two embeddings of a 2-polyhedron whose regular neighborhoods have sphere boundaries and are not homeomorphic. Thus, additional assumptions are necessary for non-prime manifolds. We make the assumption that the two embeddings are homotopic.

Theorem. Let M be a closed, orientable 3-manifold with no fake balls. Let K be a 2-polyhedron with two homotopic embeddings into M. If the boundaries of the regular neighborhoods of these embeddings are both homeomorphic to the 2-sphere, the regular neighborhoods are homeomorphic.

Thursday, July 12, 2001
DISSERTATION COMMITTEE
M. Paul Latiolais, Chair
Steven Bleiler
Andrew M. Fraser
Nancy Waller
Erik Bodegom, Graduate Studies Rep.