Semicellularity, decompositions and mappings in manifolds

Coram, Donald

doi:10.1090/S0002-9947-1974-0356068-3

Semicellularity, decompositions and mappings in manifolds
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Trans. Amer. Math. Soc. 191 (1974), 227-244 Request permission

Abstract:

If X is an arbitrary compact set in a manifold, we give algebraic criteria on X and on its embedding to determine that X has an arbitrarily small, closed neighborhood each component of which is a p-connected, piecewise linear manifold which collapses to a q-dimensional subpolyhedron from some p and q. This property generalizes cellularity. The criteria are in terms of UV properties and Alexander-Spanier cohomology. These criteria are then applied to decide when the components of a given compact set in a manifold are elements of a decomposition such that the quotient space is the n-sphere. Conversely, algebraic criteria are given for the point inverses of a map between manifolds to have arbitrarily small neighborhoods of the type mentioned above; these criteria are considerably weaker than for an arbitrary compact set.

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