In the subject of manifold theory in mathematics, if M is a topological manifold with boundary, its double is obtained by gluing two copies of M together along their common boundary. Precisely, the double is where for all. If M has a smooth structure, then its double can be endowed with a smooth structure thanks to a collar neighbourdhood. 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. 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 is. 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 is the double of M'. The double of a Mazur manifold is a homotopy 4-sphere.