Double (manifold)

For the equipment used to connect two air cylinders in SCUBA diving, see Manifold (scuba).

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

If ${\displaystyle M}$ 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 ${\displaystyle \partial M}$ is non-empty and ${\displaystyle M}$ is compact.

Doubles bound

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

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