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:
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:
[ g ] ∉ N,
g ( S 2 ) ⊂ f ( S 2 ) ∪ U,
g : S 2 → g ( S 2 )is a covering map, and
g ( S 2 )is a 2-sided submanifold (2-sphere or projective plane) of M.
quoted in Hempel (p. 54)