Sphere theorem (3-manifolds)

In mathematics, in the topology of 3-manifolds, the sphere theorem of 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.

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

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