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

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

1. $[g]\notin N$,
2. $g(S^2)\subset f(S^2)\cup U$,
3. $g:S^2\to g(S^2)$ is a covering map, and
4. $g(S^2)$ 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.