Double (manifold)

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 ${\displaystyle x\in \partial M}$. Equivalently, the double of ${\displaystyle M}$ is the boundary of ${\displaystyle M\times [0,1]}$. This gives doubles a special role in cobordism.

If ${\displaystyle M}$ has a smooth structure, then its double can be endowed with a smooth structure thanks to a collar neighbourhood.^{[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.