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 $M \times \{0,1\} / \sim$ where $(x,0) \sim (x,1)$ for all $x \in \partial M$ .

Although the concept makes sense for any manifold, the notion of double tends to be used only in the context that $\partial M$ is non-empty and $M$ is compact.

[edit] 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 $M \times D^k$ is $M \times S^k$ .

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 $M \mathrel{\#} -M$ is the double of $M'$ .