Double (manifold)

In the subject of manifold theory in mathematics, if [math]\displaystyle{ M }[/math] is a topological manifold with boundary, its double is obtained by gluing two copies of [math]\displaystyle{ M }[/math] together along their common boundary. Precisely, the double is [math]\displaystyle{ M \times \{0,1\} / \sim }[/math] where [math]\displaystyle{ (x,0) \sim (x,1) }[/math] for all [math]\displaystyle{ x \in \partial M }[/math].

If [math]\displaystyle{ M }[/math] has a smooth structure, then its double can be endowed with a smooth structure thanks to a collar neighbourdhood.^{[1]:th. 9.29 & ex. 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 [math]\displaystyle{ \partial M }[/math] is non-empty and [math]\displaystyle{ M }[/math] is compact.

Doubles bound

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

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

References

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