Sphere theorem (3-manifolds) explained

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

\pi2(M)

is not the trivial group. Then there exists a non-zero element of

\pi2(M)

having a representative that is an embedding

S2\toM

.

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

\pi1(M)

-invariant subgroup of

\pi2(M)

. If

f\colonS2\toM

is a general position map such that

[f]\notinN

and

U

is any neighborhood of the singular set

\Sigma(f)

, then there is a map

g\colonS2\toM

satisfying

[g]\notinN

,

g(S2)\subsetf(S2)\cupU

,

g\colonS2\tog(S2)

is a covering map, and

g(S2)

is a 2-sided submanifold (2-sphere or projective plane) of

M

. quoted in .

References

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