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×{0,1}/∼ where (x,0)∼(x,1) for all x∈∂M.

If 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 ∂M is non-empty and M is compact.

## Doubles bound

Given a manifold M, the double of M is the boundary of M×[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×Dk is M×Sk. 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#−M is the double of M′.

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

## References

1. ↑ Lee, John (2012), Introduction to Smooth Manifolds, Graduate Texts in Mathematics, 218, Springer, ISBN 9781441999825, https://books.google.com/books?id=xygVcKGPsNwC&pg=PA226
2. ↑ 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 . See in particular p. 24.

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