Sphere Theorem (3-Manifolds)

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 ${\displaystyle M}$ be an orientable 3-manifold such that ${\displaystyle {\pi }_2(M)}$ is not the trivial group. Then there exists a non-zero element of ${\displaystyle {\pi }_2(M)}$ having a representative that is an embedding ${\displaystyle S^2\to M}$.

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 ${\displaystyle M}$ be any 3-manifold and ${\displaystyle N}$ a ${\displaystyle {\pi }_1(M)}$-invariant subgroup of ${\displaystyle {\pi }_2(M)}$. If ${\displaystyle f:S^2\to M}$ is a general position map such that ${\displaystyle [f]\notin N}$ and ${\displaystyle U}$ is any neighborhood of the singular set ${\displaystyle \mathrm{\Sigma }(f)}$, then there is a map ${\displaystyle g:S^2\to M}$ satisfying

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

quoted in (Hempel 1976).

• 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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