Sphere theorem (3-manifolds)

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This article is about embeddings of 2-spheres. For the sphere theorem in Riemannian geometry, see Sphere theorem.

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 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 can be based on transversality methods, see Batude below.

Another more general version (also called the projective plane theorem due to 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

  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.

quoted in Hempel (p. 54)


  • Batude, J. L. (1971). "Singularité générique des applications différentiables de la 2-sphère dans une 3-variété différentiable". Annales de l'Institut Fourier 21 (3): 151–172. 
  • Hempel, J. (1978). 3-manifolds. Princeton University Press.