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Manifold Neighbourhoods and a Conjecture of Adjamagbo
Author(s) -
David Gauld
Publication year - 2021
Publication title -
new zealand journal of mathematics
Language(s) - English
Resource type - Journals
eISSN - 1179-4984
pISSN - 1171-6096
DOI - 10.53733/131
Subject(s) - submanifold , mathematics , conjecture , pure mathematics , subfamily , manifold (fluid mechanics) , neighbourhood (mathematics) , differential (mechanical device) , combinatorics , mathematical analysis , physics , mechanical engineering , biochemistry , chemistry , engineering , gene , thermodynamics
We verify a conjecture of P. Adjamagbo that if the frontier of a relatively compact subset $V_0$ of a manifold is a submanifold then there is an increasing family $\{V_r\}$ of relatively compact open sets indexed by the positive reals so that the frontier of each is a submanifold, their union is the whole manifold and for each $r\ge 0$ the subfamily indexed by $(r,\infty)$ is a neighbourhood basis of the closure of the $r^{\rm th}$ set. We use smooth collars in the differential category, regular neighbourhoods in the piecewise linear category and handlebodies in the topological category.

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