Semicellularity, decompositions and mappings in manifolds

Author:
Donald Coram

Journal:
Trans. Amer. Math. Soc. **191** (1974), 227-244

MSC:
Primary 57A60

MathSciNet review:
0356068

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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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Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9947-1974-0356068-3

Keywords:
Neighborhoods of compacta,
*UV* properties,
extending decompositions,
mappings on manifolds

Article copyright:
© Copyright 1974
American Mathematical Society