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 ${\displaystyle f\colon S^{2}\to M}$ is a general position map such that ${\displaystyle [f]\notin N}$ and ${\displaystyle U}$ is any neighborhood of the singular set ${\displaystyle \Sigma (f)}$, then there is a map ${\displaystyle g\colon S^{2}\to M}$ satisfying

1. ${\displaystyle [g]\notin N}$,
2. ${\displaystyle g(S^{2})\subset f(S^{2})\cup U}$,
3. ${\displaystyle g\colon S^{2}\to g(S^{2})}$ 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, p. 54).