Collar neighbourhood theorem

Remarks

The map $c$, or its image $U$, is a collar of $\partial M$. The idea of the proof is to choose a smooth vector field near $\partial M$ that points into $M$ along $\partial M$. Such a field is glued from local choices with a partition of unity. Its flow, started at the boundary points and run for short times, gives the collar. Example: for the closed unit disc $D^2$, the map $c(x, t) = (1 - t/2) x$ is a collar $S^1 \times [0, 1) \to D^2$ of the boundary circle. The statement also holds for topological manifolds with boundary, a result of Morton Brown.

Going deeper

For a student

Collars are used to glue manifolds along boundary components, for example in the composition of cobordisms and in handle attachments. They show that $M$ is homotopy equivalent to its interior, and that $M$ with an outer collar $\partial M \times [0, 1]$ attached is diffeomorphic to $M$. The collar theorem is the boundary analogue of the tubular neighbourhood theorem: the inward normal bundle of $\partial M$ is a trivial half-line bundle.