Double (manifold)

For the equipment used to connect two air cylinders in SCUBA diving, see Manifold (scuba).

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. Precisely, the double is $M \times \{0,1\} / \sim$ where (x,0)∼(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.

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