zbMATH — the first resource for mathematics

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)].

57N15 Topology of the Euclidean \(n\)-space, \(n\)-manifolds (\(4 \leq n \leq \infty\)) (MSC2010)
57Q99 PL-topology
Full Text: DOI
[1] Ancel, F.D.; Guilbault, C.R., Compact contractible n-manifolds have arc spines (n⩾5), Pacific J. math., 168, 1, 1-10, (1995) · Zbl 0820.57014
[2] Cohen, M.M., A course in simple-homotopy theory, (1973), Springer New York · Zbl 0261.57009
[3] Fenn, R., Embedding polyhedra, Bull. London math. soc., 2, 316-318, (1970) · Zbl 0207.53105
[4] Guilbault, C.R., Some compact contractible manifolds containing disjoint spines, Topology, 34, 1, 99-108, (1995) · Zbl 0841.57034
[5] Huck, G., Embeddings of acyclic 2-complexes in S4 with contractible complement, (), 122-129 · Zbl 0765.57004
[6] Kato, M., Embedding spheres and balls in codimension ⩽2, Invent. math., 10, 89-107, (1970) · Zbl 0201.56104
[7] Mazur, B., A note on some contractible 4-manifolds, Ann. of math. (2), 73, 221-228, (1961) · Zbl 0127.13604
[8] Newman, M.H.A., Boundaries of ulc sets in Euclidean n-space, Proc. acad. sci., 34, 193-196, (1948) · Zbl 0036.12801
[9] Rourke, C.P.; Sanderson, B.J., Introduction to piecewise-linear topology, (1982), Springer Berlin, reprint · Zbl 0477.57003
[10] Wright, D.G., Contractible open manifolds which are not covering spaces, Topology, 31, 2, 281-291, (1992) · Zbl 0755.57009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.