Sphere theorem (3-manifolds)

In the topology of 3-manifolds, the sphere theorem denotes a family of statements which show us how the image of a 2-sphere, under a continuous map into a 3-manifold, may behave.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 $S^2 o M$ .

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 $fcolon S^2 o M$ is a general position map such that $[f]
otin N$ and $U$ is any neighborhood of the singular set $Sigma(f)$ , then there is a map $gcolon S^2 o M$ satisfying

# $[g]
otin N$ ,
# $g(S^2)subset f(S^2)cup U$ ,
# $gcolon S^2 o g(S^2)$ 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)

References

* cite journal
author = Batude, J. L.
title = Singularité générique des applications différentiables de la 2-sphère dans une 3-variété différentiable
journal = Annales de l'Institut Fourier
volume = 21
issue = 3
year = 1971
pages = 151–172

* cite journal
author = Epstein, D. B. A.
authorlink = David B. A. Epstein
title = Projective planes in 3-manifolds
journal = Proceedings of the London Mathematical Society, III Ser.
volume = 11
year = 1961
pages = 469–484
doi = 10.1112/plms/s3-11.1.469

* cite book
author = Hempel, J.
title = 3-manifolds
publisher = Princeton University Press
year = 1978

* cite journal
author = C. Papakyriakopoulos
authorlink = Christos Papakyriakopoulos
title = On Dehn's lemma and asphericity of knots
journal = Annals of Mathematics
volume = 66
year = 1957
pages = 1–26
doi = 10.2307/1970113

* cite journal
author = Whitehead, J. H. C.
authorlink = J. H. C. Whitehead
title = On 2-spheres in 3-manifolds
journal = Bulletin of the American Mathematical Society
volume = 64
year = 1958
pages = 161–166
doi = 10.1090/S0002-9904-1958-10193-7

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