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 Christos 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 of the theorem can be based on transversality methods, see Jean-Loïc Batude (1971).
Another more general version (also called the projective plane theorem, and due to David B. A. 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
Batude, Jean-Loïc (1971). "Singularité générique des applications différentiables de la 2-sphère dans une 3-variété différentiable" (PDF). Annales de l'Institut Fourier. 21 (3): 151–172. doi:10.5802/aif.383. MR 0331407.
sphere, theorem, manifolds, this, article, about, embeddings, spheres, sphere, theorem, riemannian, geometry, sphere, theorem, mathematics, topology, manifolds, sphere, theorem, christos, papakyriakopoulos, 1957, gives, conditions, elements, second, homotopy, . 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 Christos 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 displaystyle M be an orientable 3 manifold such that p2 M displaystyle pi 2 M is not the trivial group Then there exists a non zero element of p2 M displaystyle pi 2 M having a representative that is an embedding S2 M displaystyle S 2 to M The proof of this version of the theorem can be based on transversality methods see Jean Loic Batude 1971 Another more general version also called the projective plane theorem and due to David B A Epstein is Let M displaystyle M be any 3 manifold and N displaystyle N a p1 M displaystyle pi 1 M invariant subgroup of p2 M displaystyle pi 2 M If f S2 M displaystyle f colon S 2 to M is a general position map such that f N displaystyle f notin N and U displaystyle U is any neighborhood of the singular set S f displaystyle Sigma f then there is a map g S2 M displaystyle g colon S 2 to M satisfying g N displaystyle g notin N g S2 f S2 U displaystyle g S 2 subset f S 2 cup U g S2 g S2 displaystyle g colon S 2 to g S 2 is a covering map and g S2 displaystyle g S 2 is a 2 sided submanifold 2 sphere or projective plane of M displaystyle M quoted in Hempel 1976 p 54 References editBatude Jean Loic 1971 Singularite generique des applications differentiables de la 2 sphere dans une 3 variete differentiable PDF Annales de l Institut Fourier 21 3 151 172 doi 10 5802 aif 383 MR 0331407 Epstein David B A 1961 Projective planes in 3 manifolds Proceedings of the London Mathematical Society 3rd ser 11 1 469 484 doi 10 1112 plms s3 11 1 469 Hempel John 1976 3 manifolds Annals of Mathematics Studies Vol 86 Princeton NJ Princeton University Press MR 0415619 Papakyriakopoulos Christos 1957 On Dehn s lemma and asphericity of knots Annals of Mathematics 66 1 1 26 doi 10 2307 1970113 JSTOR 1970113 PMC 528404 Whitehead J H C 1958 On 2 spheres in 3 manifolds Bulletin of the American Mathematical Society 64 4 161 166 doi 10 1090 S0002 9904 1958 10193 7 Retrieved from https en wikipedia org w index php title Sphere theorem 3 manifolds amp oldid 1142493145, wikipedia, wiki, book, books, library,
