Double (manifold)
In the subject of manifold theory in mathematics, if is a manifold with boundary, its double is obtained by gluing two copies of together along their common boundary. Precisely, the double is where for all.
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 is non-empty and is compact.Given a manifold, the double of 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 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 is a closed, oriented manifold and if is obtained from by removing an open ball, then the connected sum is the double of.
The double of a Mazur manifold is a homotopy 4-sphere.
