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The approximate tubular neighborhood theorem. (English) Zbl 1030.57034

For smooth manifolds, the classical theorem of Whitney asserts the existence of a tubular neighbourhood to any smooth submanifold. In the case of topological manifolds, only weaker results hold. If one considers topologically stratified spaces, skeleton and strata may fail to have mapping cylinder neighbourhoods. In the paper under review, it is shown that under some dimension and compactness assumptions pure subsets of manifold stratified spaces have approximate tubular neighborhoods, these ones being generalizations of mapping cylinder ones. The generalization is necessary, thus this is the best possible topological tubular neighbourhood theorem in the stratified setting.

MSC:

57N80 Stratifications in topological manifolds
57N40 Neighborhoods of submanifolds
55R65 Generalizations of fiber spaces and bundles in algebraic topology
58A35 Stratified sets
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