Английская Википедия:Double (manifold)

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

If <math>M</math> has a smooth structure, then its double can be endowed with a smooth structure thanks to a collar neighbourdhood.Шаблон:Rp

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>\partial M</math> is non-empty and <math>M</math> is compact.

Doubles bound

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

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