In mathematics, in the topology of 3manifolds, the sphere theorem of Papakyriakopoulos (1957) gives conditions for elements of the second homotopy group of a 3manifold to be represented by embedded spheres.
One example is the following:
Let
M
be an orientable 3manifold such that
π
2
(
M
)
is not the trivial group. Then there exists a nonzero element of
π
2
(
M
)
having a representative that is an embedding
S
2
→
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 3manifold and
N
a
π
1
(
M
)
invariant subgroup of
π
2
(
M
)
. If
f
:
S
2
→
M
is a general position map such that
[
f
]
∉
N
and
U
is any neighborhood of the singular set
Σ
(
f
)
, then there is a map
g
:
S
2
→
M
satisfying

[
g
]
∉
N
,

g
(
S
2
)
⊂
f
(
S
2
)
∪
U
,

g
:
S
2
→
g
(
S
2
)
is a covering map, and

g
(
S
2
)
is a 2sided submanifold (2sphere or projective plane) of
M
.
quoted in Hempel (p. 54)