Handlebody Neighbourhoods and a conjecture of Adjamagbo

TL;DR

This paper proves a conjecture by Adjamagbo using handlebodies, showing that certain submanifolds can be extended to a nested family covering the entire manifold with specific boundary properties.

Contribution

It introduces a handlebody-based method to verify a conjecture about submanifold frontiers and nested neighborhoods in manifolds.

Findings

Verified Adjamagbo's conjecture for manifolds with handlebody techniques.

Constructed an increasing family of open sets with submanifold frontiers.

Established a neighborhood basis for the manifold's closure.

Abstract

Using handlebodies 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.