nLab collar neighbourhood theorem




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.


Let XX be a topological manifold or smooth manifold with boundary X\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 X×[0,1)\partial X \times [0,1) of X\partial X with the half-open interval:

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


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.