In mathematics, in the topology of 3-manifolds, the sphere theorem of 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 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 .
The proof of this version can be based on transversality methods, see Batude below.
Another more general version (also called the projective plane theorem due to Epstein) 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
quoted in Hempel (p. 54)