Manifolds accepting codimension-one-sphere-shape decompositions (Q1068387)

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 \textit{L. C. Siebenmann} [Comment. Math. Helv. 45, 1-48 (1970; Zbl 0215.246)] and \textit{W. Browder} and \textit{T. Levine} [ibid. 40, 153-160 (1966; Zbl 0134.428)].