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, MR 780575, doi:10.1090/conm/035/780575. See in particular p. 24.
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