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Double (manifold)

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In the subject of manifold theory in mathematics, if is a topological manifold with boundary, its double is obtained by gluing two copies of together along their common boundary. Precisely, the double is where for all .

If 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 is non-empty and is compact.

Doubles bound

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Given a manifold , the double of is the boundary of . This gives doubles a special role in cobordism.

Examples

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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

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  1. ^ Lee, John (2012), Introduction to Smooth Manifolds, Graduate Texts in Mathematics, vol. 218, Springer, ISBN 9781441999825
  2. ^ Aitchison, I. R.; Rubinstein, J. H. (1984), "Fibered knots and involutions on homotopy spheres", Four-manifold theory (Durham, N.H., 1982), Contemp. Math., vol. 35, Amer. Math. Soc., Providence, RI, pp. 1–74, doi:10.1090/conm/035/780575, MR 0780575. See in particular p. 24.