Disjoint Spines in Newman Contractible Manifolds
by
Manuel (Bud) Sanders
University of Tennessee

A compact polyhedron K in the interior of a PL manifold M is said to be a spine of M if M collapses to K. The manifold M has disjoint spines provided that it collapses (independently) to two disjoint polyhedra in its interior. The question arises as to which contractible manifolds have a pair of disjoint spines. A technique of M.H.A. Newman provides contractible manifolds which are not balls. A Newman contractible manifold is constructed as the closure of the complement of a regular neighborhood of a finite, acyclic, simplicial complex K in Sⁿ with \pi₁(K) =/= { 1 } for large enough n. When well-defined, it is denoted New(K, n). C. Guilbault has shown that if K is any finite, non-simply connected, acyclic k-complex then New(K, n) has disjoint spines provided n > 4k thereby providing examples in dimensions n >= 9. Some results concerning disjoint spines in lower dimensional Newman contractible manifolds will be discussed.

Date received: May 22, 1998