# Double (manifold)

In the subject of manifold theory in mathematics, if is a manifold with boundary, its **double** is obtained by gluing two copies of together along their common boundary.^{[1]} Precisely, the double is where for all .

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 is non-empty and is compact.

## Doubles bound

Given a manifold , the **double** of is the boundary of . 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 is closed, the double of is . 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 is a closed, oriented manifold and if is obtained from by removing an open ball, then the connected sum is the double of .

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

## References

- ↑ Lee, John (2012),
*Introduction to Smooth Manifolds*, Graduate Texts in Mathematics,**218**, Springer, p. 226, ISBN 9781441999825. - ↑ Aitchison, I. R.; Rubinstein, J. H. (1984), "Fibered knots and involutions on homotopy spheres",
*Four-manifold theory (Durham, N.H., 1982)*, Contemp. Math.,**35**, Amer. Math. Soc., Providence, RI, pp. 1–74, doi:10.1090/conm/035/780575, MR 780575. See in particular p. 24.