12

The well-known collar neighbourhood theorem states:

Let $M$ be a smooth manifold with compact boundary $\partial M$, then there exists a neighbourhood of $\partial M$, which is diffeomorphic to $\partial M\times [0,1)$.

I am asking myself if the theorem holds in the topological category. Is it true at least for compact topological manifolds?

My first idea would be to take a more closer look at the work of Kirby and Siebenmann on topological manifolds, but since I am absolutely not an expert in this field, I hoped to get an answer with a reference or a counterexample here.

archipelago
  • 4,478

1 Answers1

12

This is a special case of the main result of this paper:

Morton Brown, "Locally flat imbeddings of topological manifolds", Annals of Mathematics, Vol. 75 (1962), p. 331-341.

Edit: See also Proposition 3.42 in Hatcher's "Algebraic Topology" for a self-contained proof of the existence of a boundary collar for manifolds with boundary. Hatcher assumes that the manifold is compact but his proof only uses compactness of the boundary.

Moishe Kohan
  • 114,140
  • Does the proof require us to think of the boundary as an embedded submanifold? My naive understanding is that the boundary of a manifold $M$ can be embedded in $M$ if it is endowed with subspace topology. However, this does not seem to be necessary (or is it?!) and the boundary, as a manifold by itself, can have a different topology than the subspace topology. If it has a different topology, it is not necessarily an embedded submanifold of $M$ (is it still a submanifold?!). In that situation, does the collar neighbourhood theorem still hold? – QGravity Apr 04 '24 at 22:31
  • 1
    @QGravity: The subspace topology is the default one. – Moishe Kohan Apr 04 '24 at 22:53