Double (manifold)

In the subject of manifold theory in mathematics, if $M$ is a manifold with boundary, its double is obtained by gluing two copies of $M$ together along their common boundary.^[1] Precisely, the double is $M \times \{0,1\} / \sim$ where $(x,0) \sim (x,1)$ for all $x \in \partial M$.

Although the concept makes sense for any manifold, and even for some non-manifold sets such as the Alexander horned sphere, the notion of double tends to be used primarily in the context that $\partial M$ is non-empty and $M$ is compact.

Doubles bound

Given a manifold $M$, the double of $M$ is the boundary of $M \times [0,1]$. This gives doubles a special role in cobordism.

Examples

The n-sphere is the double of the n-ball. In this context, the two balls would be the upper and lower hemi-sphere respectively. More generally, if $M$ is closed, the double of $M \times D^k$ is $M \times S^k$. Even more generally, the double of a disc bundle over a manifold is a sphere bundle over the same manifold. More concretely, the double of the Möbius strip is the Klein bottle.

If $M$ is a closed, oriented manifold and if $M'$ is obtained from $M$ by removing an open ball, then the connected sum $M \mathrel{\#} -M$ is the double of $M'$.

The double of a Mazur manifold is a homotopy 4-sphere.^[2]

References

1. Lee, John (2012), Introduction to Smooth Manifolds, Graduate Texts in Mathematics 218, Springer, p. 226, ISBN 9781441999825.
2. Aitchison, I. R.; Rubinstein, J. H. (1984), "Fibered knots and involutions on homotopy spheres", Four-manifold theory (Durham, N.H., 1982), Contemp. Math. 35, Amer. Math. Soc., Providence, RI, pp. 1–74, doi:10.1090/conm/035/780575, MR 780575. See in particular p. 24.