Canonical neighborhoods for topologically embedded polyhedra

Craggs, Robert

doi:10.1090/S0002-9947-1972-0394687-7

Canonical neighborhoods for topologically embedded polyhedra
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Trans. Amer. Math. Soc. 173 (1972), 465-490 Request permission

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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