Existence of certain compact contractible manifolds containing disjoint spines

Author

Manuel Jackson Sanders

Date of Award

5-1999

Degree Type

Dissertation

Degree Name

Doctor of Philosophy

Major

Mathematics

Major Professor

Robert J. Daverman

Committee Members

Jerzy Dydak, Lawrence Husch, Klaus Johannson, Jeff Kovac

Abstract

A polyhedron K in the interior of a compact PL manifold M (with boundary) is said to be a (PL) spine of M provided M collapses to K. The manifold M has disjoint spines provided it collapses (independently) to two disjoint such polyhedra in its interior. A long-standing conjecture asserts that a certain class of compact, contractible 4-manifolds constructed by Mazur do not have disjoint spines. More recently, the question as to which compact, contractible manifolds have a pair of disjoint spines has been expanded by C. Guilbault so as to include other compact, contractible manifolds, including those in higher dimensions.

A technique of M.H.A. Newman provides compact, contractible manifolds which are not balls. A Newman manifold is constructed as the closure of the complement of a regular neighborhood of a non-simply connected, finite, acyclic, simplicial complex. K, in Sⁿ for sufficiently large n. When well-defined, it is denoted New(K, n). Guilbault has shown that if K is a non-simply connected, finite, acyclic, simplicial k-complex, then, New(K, n) has disjoint spines provided n > 4k. His techniques, thereby, provide interesting examples of disjoint spine phenomena in dimensions n ≥ 9 and are the only known examples of such occurrences.

In this work, it is shown that the result holds in dimension n = 4k as well. Moreover, if K is a non-simply connected, finite, acyclic 2-complex, it is shown that New(K, n) has disjoint spines if n ≥ 5, thereby producing relatively low-dimensional examples of the phenomenon. Since it is known that no interesting examples can occur in dimensions n ≤ 3, these results show that the question regarding existence of interesting examples is, now, undecided only in dimension 4.

Recommended Citation

Sanders, Manuel Jackson, "Existence of certain compact contractible manifolds containing disjoint spines. " PhD diss., University of Tennessee, 1999.
https://trace.tennessee.edu/utk_graddiss/8919