Collar neighborhood theorem - Imported Knowlpedia Content

Statement

Let $M$ be a smooth manifold with boundary. There exists a smooth embedding

c:\partial M\times[0,1)\longrightarrow M

such that $c(x,0)=x$ for every $x\in\partial M$ , and whose image is an open neighborhood of $\partial M$ in $M$ . Such an embedding is called a collar of the boundary. Equivalently, $c$ is a diffeomorphism from the product manifold $\partial M\times[0,1)$ onto a neighborhood of the boundary, relative to the fixed boundary inclusion. No compactness assumption on $M$ or $\partial M$ is required Lee, Chapter 9. Here smoothness is understood in the manifold-with-boundary sense, including extension across the endpoint.

Construction idea

Choose a smooth vector field along the boundary that points inward and extend it to a neighborhood. Its local flow carries each boundary point into the interior. Flow existence gives a product map for a point-dependent time; a positive rescaling of the vector field and a locally finite construction make the interval uniform. Injectivity after shrinking produces the collar.

Consequences

A collar gives a boundary defining function by taking the interval coordinate near $\partial M$ and extending it positively over the rest of $M$ . It also provides room for gluing manifolds along diffeomorphic boundary components: product coordinates identify smooth structures on the two sides. In cobordism, collars ensure that composition by boundary gluing is independent of accidental coordinate behavior at the seam.

The inclusion $\partial M\hookrightarrow c(\partial M\times[0,1))$ is a deformation retract, with homotopy $c(x,t)\mapsto c(x,(1-s)t)$ . This conclusion concerns the collar neighborhood, not generally the whole manifold.

Examples and nonuniqueness

For $M=N\times[0,1)$ , the identity map is the standard collar. A closed interval has a collar consisting of two disjoint short half-intervals, one at each endpoint. On the closed ball, radial motion inward gives a collar of the sphere.

Collars are not canonical: different inward vector fields give different embeddings. The map $c(x,t)=c_0(x,t^2)$ built from a collar $c_0$ is not another collar, because its differential loses the normal direction at $t=0$ .

References

John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Springer, 2013. Springer DOI record. Relevant: Chapter 9, boundary flowouts and the collar neighborhood theorem.
Morris W. Hirsch, Differential Topology, Springer, 1976. Springer DOI record. Relevant: Chapter 4, collars and tubular-neighborhood methods.