Shrinking cell-like decompositions of manifolds. Codimension three

Euclidean n-space En, n > 5, has the following simple DISJOINT DISK PROPERTY: singular 2-dimensional disks in En may be adjusted slightly so as to be disjoint. We show that for a large class of cell-like decompositions of manifolds this property in the decomposition space is sufficient in order that the decomposition space be a manifold. As a consequence we deduce the DOUBLE SUSPENSION THEOREM proved in a large number of cases by R. D. Edwards: The double suspension of any homology sphere is a topological sphere. We also obtain a sweeping generalization of Edwards' MANIFOLD FACTOR THEOREM; Edwards' theorem states that, if X is a single cell-like set in Euclidean n-dimensional space En, then (En/X) x E = En+l.

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