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Manifolds accepting codimension-one-sphere-shape decompositions. (English) Zbl 0582.57009

This paper is concerned with the following question. Which connected closed m-manifolds can accept decompositions so that each element of the decomposition is shape equivalent to the (m-1)-sphere \(S^{m-1} ?\) The author shows that all such decompositions must be upper semicontinuous for \(m\geq 2\). He then shows for \(m\neq 3,4,5\) that a manifold accepting such a decomposition must be homeomorphic to the total space of an \(S^{m-1}\)-fiber bundle over the circle. The paper relies on results about fibering over the circle by L. C. Siebenmann [Comment. Math. Helv. 45, 1-48 (1970; Zbl 0215.246)] and W. Browder and T. Levine [ibid. 40, 153-160 (1966; Zbl 0134.428)].
Reviewer: D.G.Wright

MSC:

57N15 Topology of the Euclidean \(n\)-space, \(n\)-manifolds (\(4 \leq n \leq \infty\)) (MSC2010)
57R22 Topology of vector bundles and fiber bundles
54B15 Quotient spaces, decompositions in general topology
55R10 Fiber bundles in algebraic topology
55P55 Shape theory
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