“Sphere theorem (3-manifolds)”的意思、由来-开放百科全书

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Sphere theorem (3-manifolds)

释义

References

{{about|embeddings of 2-spheres|the sphere theorem in Riemannian geometry|Sphere theorem}}

In mathematics, in the topology of 3-manifolds, the sphere theorem of {{harvs|txt|last=Papakyriakopoulos|first=Christos|authorlink=Christos Papakyriakopoulos|year=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 be an orientable 3-manifold such that is not the trivial group. Then there exists a non-zero element of having a representative that is an embedding .

The proof of this version of the theorem can be based on transversality methods, see {{harvs|txt|last=Batude|first=Jean-Loïc|year=1971}}.

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

Let be any 3-manifold and a -invariant subgroup of . If is a general position map such that and is any neighborhood of the singular set , then there is a map satisfying

,
,
is a covering map, and
is a 2-sided submanifold (2-sphere or projective plane) of .

quoted in {{harv|Hempel|year=1978|p=54}}.

References

{{cite journal

{{cite journal

| last = Epstein|first= David B. A.
| authorlink = David B. A. Epstein
| title = Projective planes in 3-manifolds
| journal = Proceedings of the London Mathematical Society |series=3rd ser.
| volume = 11
| year = 1961
| issue = 1
| pages = 469–484
| doi = 10.1112/plms/s3-11.1.469}}

{{cite book

| last = Hempel|first=John
| title = 3-manifolds
| publisher = Princeton University Press
| location = Princeton, NJ
|series= Annals of Mathematics Studies| volume=86
| mr=0415619
| year = 1976}}

{{cite journal

{{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}}

3 : Geometric topology|3-manifolds|Theorems in topology

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