nLab collar neighbourhood theorem

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:

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

Related concepts

• cobordism

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)