Abstract
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.
Highlights
Pascal Adjamagbo, [1], has proposed the following conjecture1: Given a relatively compact non-empty open subset V0 of a connected manifold M m such that the boundary of V0 is a submanifold, there exists an increasing family Vr r∈[0,∞) of relatively compact open subsets of M the boundaries of which are submanifolds such that M is the union of the
Elements of the family, and that for any r ∈ [0, ∞), the family Vs s>r is a fundamental system of neighbourhoods of the closure of Vr
Adjamagbo makes no assumptions regarding the tameness of the boundary manifold, so it could be wild at every point; see [6, Theorem 2.6.1] for example
Summary
Pascal Adjamagbo, [1], has proposed the following conjecture: Given a relatively compact non-empty open subset V0 of a connected manifold M m such that the boundary of V0 is a submanifold, there exists an increasing family Vr r∈[0,∞) of relatively compact open subsets of M the boundaries of which are submanifolds such that M is the union of the. Adjamagbo makes no assumptions regarding the tameness of the boundary manifold, so it could be wild at every point; see [6, Theorem 2.6.1] for example. In both the piecewise linear and topological categories we do not need the boundary of V0 to be a manifold for all of the rest of the conjecture to be satisfied. The differential category is easier to deal with than the topological and piecewise linear categories but it uses a technique that helps us in the other two Any neighbourhood of V0 must contain the set (−ε, 0) × N/ ∼ for some ε > 0 and its closure contains all of the infinitely many ‘origins’ (0, n) so cannot be compact
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
