Sphere Theorem (3-manifolds)

In mathematics, in the topology of 3-manifolds, the sphere theorem of 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 of the theorem can be based on transversality methods, see .

Another more general version (also called the projective plane theorem, and due to David B. A. 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 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

# notin N,
#$g(S^2)\subset f(S^2)\cup U$,
#$g\colon S^2\to g(S^2)$ is a covering map, and
#$g(S^2)$ is a 2-sided submanifold (2-sphere or projective plane) of $M$.

quoted in .

References

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* {{cite journal
, last = Whitehead, first= 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
, issue = 4
, pages = 161–166
, doi = 10.1090/S0002-9904-1958-10193-7, doi-access = free

Geometric topology
3-manifolds
Theorems in topology