Sphere theorem (3-manifolds)

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This article is about embeddings of 2-spheres. For the sphere theorem in Riemannian geometry, see Sphere theorem.

In mathematics, in the topology of 3-manifolds, the sphere theorem of Papakyriakopoulos (1957) gives conditions for elements of the second homotopy group of a 3-manifold to be represented by embedded spheres.

One example is the following:

Let be an orientable 3-manifold such that is not the trivial group. Then there exists a non-zero element of having a representative that is an embedding .

The proof of this version can be based on transversality methods, see Batude below.

Another more general version (also called the projective plane theorem due to Epstein) is:

Let be any 3-manifold and a -invariant subgroup of . If is a general position map such that and is any neighborhood of the singular set , then there is a map satisfying

  1. ,
  2. ,
  3. is a covering map, and
  4. is a 2-sided submanifold (2-sphere or projective plane) of .

quoted in Hempel (p. 54)

References[edit]

  • Batude, J. L. (1971). "Singularité générique des applications différentiables de la 2-sphère dans une 3-variété différentiable". Annales de l'Institut Fourier. 21 (3): 151–172. 
  • Hempel, J. (1978). 3-manifolds. Princeton University Press.