In the subject of manifold theory in mathematics, if M {\displaystyle M} is a topological manifold with boundary, its double is obtained by gluing two copies of M {\displaystyle M} together along their common boundary. Precisely, the double is M × { 0 , 1 } / ∼ {\displaystyle M\times \{0,1\}/\sim } where ( x , 0 ) ∼ ( x , 1 ) {\displaystyle (x,0)\sim (x,1)} for all x ∈ ∂ M {\displaystyle x\in \partial M} . Equivalently, the double of M {\displaystyle M} is the boundary of M × [ 0 , 1 ] {\displaystyle M\times [0,1]} . This gives doubles a special role in cobordism. If M {\displaystyle M} has a smooth structure, then its double can be endowed with a smooth structure thanks to a collar neighbourhood. 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 ∂ M {\displaystyle \partial M} is non-empty and M {\displaystyle M} is compact.
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 {\displaystyle M} is closed, the double of M × D k {\displaystyle M\times D^{k}} is M × S k {\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 M {\displaystyle M} is a closed, oriented manifold and if M ′ {\displaystyle M'} is obtained from M {\displaystyle M} by removing an open ball, then the connected sum M # − M {\displaystyle M{\mathrel {\#}}-M} is the double of M ′ {\displaystyle M'} . The double of a Mazur manifold is a homotopy 4-sphere.
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