Sphere theorem (3-manifolds)

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

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 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 {\displaystyle M} be an orientable 3-manifold such that  π 2 ( M ) {\displaystyle \pi _{2}(M)} is not the trivial group. Then there exists a non-zero element of  π 2 ( M ) {\displaystyle \pi _{2}(M)} having a representative that is an embedding S 2 -> M {\displaystyle S^{2}\to 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 {\displaystyle M} be any 3-manifold and  N {\displaystyle N} a  π 1 ( M ) {\displaystyle \pi _{1}(M)} -invariant subgroup of  π 2 ( M ) {\displaystyle \pi _{2}(M)} . If  f : S 2 -> M {\displaystyle f\colon S^{2}\to M} is a general position map such that  [ f ] ∉ N {\displaystyle [f]\notin N} and  U {\displaystyle U} is any neighborhood of the singular set  Σ ( f ) {\displaystyle \Sigma (f)} , then there is a map  g : S 2 -> M {\displaystyle g\colon S^{2}\to M} satisfying

1. [ g ] ∉ N {\displaystyle [g]\notin N} ,
2. g ( S 2 ) ⊂ f ( S 2 ) ∪ U {\displaystyle g(S^{2})\subset f(S^{2})\cup U} ,
3. g : S 2 -> g ( S 2 ) {\displaystyle g\colon S^{2}\to g(S^{2})} is a covering map, and
4. g ( S 2 ) {\displaystyle g(S^{2})} is a 2-sided submanifold (2-sphere or projective plane) of  M {\displaystyle M} .

quoted in (Hempel 1976, p. 54).

## References

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* Batude, Jean-Loïc (1971). "Singularité générique des applications différentiables de la 2-sphère dans une 3-variété différentiable" (PDF). Annales de l'Institut Fourier. 21 (3): 151–172\. doi:10.5802/aif.383. MR 0331407.

* 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. Vol. 86\. Princeton, NJ: Princeton University Press. MR 0415619.

* Papakyriakopoulos, Christos (1957). "On Dehn's lemma and asphericity of knots". Annals of Mathematics. 66 (1): 1–26\. doi:10.2307/1970113. JSTOR 1970113. PMC 528404.

* 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.