Double (manifold)

In the subject of manifold theory in mathematics, if $$M$$ is a topological 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 $$x \in \partial M$$.

If $$M$$ has a smooth structure, then its double can be endowed with a smooth structure thanks to a collar neighbourdhood.

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 $$\partial M$$ is non-empty and $$M$$ is compact.

Doubles bound
Given a manifold $$M$$, the double of $$M$$ is the boundary of $$M \times [0,1]$$. 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 $$M$$ is closed, the double of $$M \times D^k$$ is $$M \times S^k$$. 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 $$M$$ 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 $$M'$$.

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