In the subject of manifold theory in mathematics, if
is a topological manifold with boundary, its double is obtained by gluing two copies of
has a smooth structure, then its double can be endowed with a smooth structure thanks to a collar neighbourdhood.[1]: th.
9.32 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
This gives doubles a special role in cobordism.
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.
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.
is a closed, oriented manifold and if
by removing an open ball, then the connected sum
The double of a Mazur manifold is a homotopy 4-sphere.
[2] This topology-related article is a stub.