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Disjoint spine phenomena in certain contractible \(n\)-manifolds (\(n\geq 5\)). (English) Zbl 0998.57044
The present paper takes into account the problem of the existence of non trivial, compact, contractible PL-manifolds having disjoint spines; recall that this problem was originally settled, in dimension four, in connection with covering properties of Mazur manifold [D. G. Wright, Topology 31, No. 2, 281-291 (1992; Zbl 0755.57009)].
By improving techniques of [C. R. Guilbault, Topology 34, No. 1, 99-108 (1995; Zbl 0841.57034)] (which positively solves the problem in dimension \(n \geq 9\)) and by making use of properties about vanishing relative homotopy groups and of a suitable construction on acyclic 2-complexes, the author produces contractible manifolds with disjoint spines in dimension \(n \geq 5\).
Thus, the problem results to be unsolved only in the original case of dimension four.


Finally, note that both in Guilbault’s work and in the present work, all provided examples belong to the class of Newman Contractible Manifolds [M. H. A. Newman, Proc. Natl. Acad. Sci. USA 34, 193-196 (1948; Zbl 0036.12801)].

MSC:
57N15 Topology of the Euclidean \(n\)-space, \(n\)-manifolds (\(4 \leq n \leq \infty\)) (MSC2010)
57Q99 PL-topology
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