manifolds and cobordisms
cobordism theory, Introduction
Definitions
Genera and invariants
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Theorems
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 $X$ 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:
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
Created on June 17, 2019 at 11:33:33. See the history of this page for a list of all contributions to it.