Existence of certain compact contractible manifolds containing disjoint spines

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.