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 be an orientable 3-manifold such that is not the trivial group. Then there exists a non-zero element of having a representative that is an embedding. This statement may be strengthened to show that the embedding is piecewise linear.
The proof of this version of the theorem can be based on transversality methods, see.
Another more general version is:
Let be any 3-manifold and a -invariant subgroup of. If is a general position map such that and is any neighborhood of the singular set, then there is a map satisfying