Sphere theorem (3-manifolds)

Short description: On when elements of the 2nd homotopy group of a 3-manifold can be embedded spheres

In mathematics, in the topology of 3-manifolds, the sphere theorem of Christos 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 M be an orientable 3-manifold such that π2(M) is not the trivial group. Then there exists a non-zero element of π2(M) having a representative that is an embedding S2->M.

The proof of this version of the theorem can be based on transversality methods, see Jean-Loïc Batude (1971).

Another more general version (also called the projective plane theorem, and due to David B. A. Epstein) is:

Let M be any 3-manifold and N a π1(M)-invariant subgroup of π2(M). If f:S2->M is a general position map such that [f]∉N and U is any neighborhood of the singular set Σ(f), then there is a map g:S2->M satisfying

1. [g]∉N,
2. g(S2)⊂f(S2)∪U,
3. g:S2->g(S2) is a covering map, and
4. g(S2) is a 2-sided submanifold (2-sphere or projective plane) of M.

quoted in (Hempel 1976).

## References

* Batude, Jean-Loïc (1971). "Singularité générique des applications différentiables de la 2-sphère dans une 3-variété différentiable". Annales de l'Institut Fourier 21 (3): 151–172. doi:10.5802/aif.383. http://www.numdam.org/article/AIF_1971__21_3_155_0.pdf.

* Epstein, David B. A. (1961). "Projective planes in 3-manifolds". Proceedings of the London Mathematical Society. 3rd ser. 11 (1): 469–484. doi:10.1112/plms/s3-11.1.469.

* Hempel, John (1976). 3-manifolds. Annals of Mathematics Studies. 86. Princeton, NJ: Princeton University Press.

* Papakyriakopoulos, Christos (1957). "On Dehn's lemma and asphericity of knots". Annals of Mathematics 66 (1): 1–26. doi:10.2307/1970113.

* Whitehead, J. H. C. (1958). "On 2-spheres in 3-manifolds". Bulletin of the American Mathematical Society 64 (4): 161–166. doi:10.1090/S0002-9904-1958-10193-7.

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