Contents

# Contents

## Idea

The collar neighbourhood theorem due to Brown 62, hence also known as Brown’s collaring theorem, says that the boundary of any manifold with boundary always admits a “collar”, namely an open neighbourhood which is the Cartesian product of the boundary with a half-open interval.

This theorem is central notably for the definition and behaviour of categories of cobordisms.

## Statement

Let $X$ be a topological manifold or smooth manifold with boundary $\partial X$. Then the boundary subspace inclusion has an open neighbourhood which is homeomorphic or diffeomorphic, respectively, to a collar, i.e. to the Cartesian product manifold with boundary $\partial X \times [0,1)$ of $\partial X$ with the half-open interval:

$\partial X \overset{ (id, 0) }{\hookrightarrow} (\partial X) \times [0,1) \overset{et}{\hookrightarrow} X \,.$

## References

Original references are:

• Morton Brown, Locally flat imbeddings of topological manifolds, Annals of Mathematics, Vol. 75 (1962), p. 331-341 (jstor:1970177)

• Robert Connelly, A new proof of Brown’s collaring theorem, Proceedings of the American Mathematical Society 27 (1971), 180 – 182 (jstor:2037284)

Quick review and sketch of the proof is in

• p. 5 of Manifolds with boundary (pdf, pdf)

Created on June 17, 2019 at 11:33:33. See the history of this page for a list of all contributions to it.