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
π
2
(
M
)
{\displaystyle \pi _{2}(M)}
is not the trivial group. Then there exists a non-zero element of
π
2
(
M
)
{\displaystyle \pi _{2}(M)}
having a representative that is an embedding
S
2
→
M
{\displaystyle S^{2}\to M}
.
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
M
{\displaystyle M}
be any 3-manifold and
N
{\displaystyle N}
a
π
1
(
M
)
{\displaystyle \pi _{1}(M)}
-invariant subgroup of
π
2
(
M
)
{\displaystyle \pi _{2}(M)}
. If
f
:
S
2
→
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
Σ
(
f
)
{\displaystyle \Sigma (f)}
, then there is a map
g
:
S
2
→
M
{\displaystyle g\colon S^{2}\to M}
satisfying
[
g
]
∉
N
{\displaystyle [g]\notin N}
,
g
(
S
2
)
⊂
f
(
S
2
)
∪
U
{\displaystyle g(S^{2})\subset f(S^{2})\cup U}
,
g
:
S
2
→
g
(
S
2
)
{\displaystyle g\colon S^{2}\to g(S^{2})}
is a covering map, and
g
(
S
2
)
{\displaystyle g(S^{2})}
is a 2-sided submanifold (2-sphere or projective plane) of
M
{\displaystyle M}
.
quoted in (Hempel 1976, p. 54).
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