Given a manifold ${\displaystyle M}$ , the double of ${\displaystyle M}$  is the boundary of ${\displaystyle M\times [0,1]}$ . 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. More generally, if ${\displaystyle M}$  is closed, the double of ${\displaystyle M\times D^{k}}$  is ${\displaystyle M\times S^{k}}$ . 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 ${\displaystyle M}$  is a closed, oriented manifold and if ${\displaystyle M'}$  is obtained from ${\displaystyle M}$  by removing an open ball, then the connected sum ${\displaystyle M{\mathrel {\#}}-M}$  is the double of ${\displaystyle M'}$ .

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