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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Canonical neighborhoods for topologically embedded polyhedra

Author: Robert Craggs
Journal: Trans. Amer. Math. Soc. 173 (1972), 465-490
MSC: Primary 57C40; Secondary 57A10
MathSciNet review: 0394687
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Abstract: D. R. McMillan has shown that in any neighborhood of a compact two sided surface in a $ 3$-manifold there is a closed neighborhood of the surface which is the sum of a solid homeomorphic to the cartesian product of the surface with the unit interval and some small disjoint cubes-with-handles each of which intersects the cartesian product in a disk on its boundary. In the present paper the author generalizes this notion of canonical neighborhood so that it applies to topological embeddings of arbitrary polyhedra in $ 3$-manifolds. This is done by replacing the cartesian products by small regular neighborhoods of polyhedral approximations to the topological embeddings.

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Keywords: $ 3$-manifold, neighborhoods of surfaces, canonical neighborhoods, topological embeddings of polyhedra, small regular neighborhoods, approximating polyhedra, cubes-with-handles
Article copyright: © Copyright 1972 American Mathematical Society