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Double (manifold)

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In the subject of manifold theory in mathematics, if M is a manifold with boundary, its double is obtained by gluing two copies of M together along their common boundary. Precisely, the double is M × { 0 , 1 } / ∼ where ( x , 0 ) ∼ ( x , 1 ) for all x ∈ ∂ M .

Contents

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 is non-empty and M is compact.

Doubles bound

Given a manifold M , the double of M is the boundary of M × [ 0 , 1 ] . 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 M is closed, the double of M × D k is M × 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 is a closed, oriented manifold and if M ′ is obtained from M by removing an open ball, then the connected sum M # − M is the double of M ′ .

The double of a Mazur manifold is a homotopy 4-sphere.

References

Double (manifold) Wikipedia
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