About: Sphere theorem (3-manifolds)

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In mathematics, in the topology of 3-manifolds, the sphere theorem of Christos Papakyriakopoulos 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 Jean-Loïc Batude. quoted in .

Attributes	Values
rdf:type	yago:WikicatTheoremsInTopology yago:Abstraction100002137 yago:Communication100033020 yago:Message106598915 yago:Proposition106750804 yago:Statement106722453 yago:Theorem106752293
rdfs:label	Sphärensatz (Topologie) (de) Sphere theorem (3-manifolds) (en) Теорема о сфере (трёхмерная топология) (ru)
rdfs:comment	In der Topologie, einem Teilgebiet der Mathematik, ist der Sphärensatz ein grundlegender Lehrsatz aus der Theorie 3-dimensionaler Mannigfaltigkeiten. Er wurde 1957 von Christos Papakyriakopoulos bewiesen. Ebenso wie der unter dem Namen Dehns Lemma bekannte Schleifensatz stellt er einen Zusammenhang zwischen der (in algebraischen Begriffen formulierbaren) Homotopietheorie und der geometrischen Topologie von 3-Mannigfaltigkeiten her; beide Sätze bilden die Grundlage für große Teile der Theorie der 3-Mannigfaltigkeiten. (de) Теорема о сфере — классическое утверждение трёхмерной топологии, доказанное Христосом Папакирьякопулосом в 1956 году вместе с леммой Дена и теоремой о петле. (ru) In mathematics, in the topology of 3-manifolds, the sphere theorem of Christos Papakyriakopoulos 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 Jean-Loïc Batude. quoted in . (en)
dcterms:subject	3-manifolds Theorems in topology Geometric topology
Wikipage page ID	4375347 (xsd:integer)
Wikipage revision ID	1021893719 (xsd:integer)
Link from a Wikipage to another Wikipage	Princeton University Press Projective plane Proceedings of the London Mathematical Society David B. A. Epstein Annales de l'Institut Fourier Annals of Mathematics Invariant (mathematics) 3-manifolds Theorems in topology General position Embedding 2-sphere Topology 3-manifold 2-sided Geometric topology Bulletin of the AMS Transversality theorem Orientable Covering map
Link from a Wikipage to an external page	http://www.numdam.org/article/AIF_1971__21_3_155_0.pdf
sameAs	Sphere theorem (3-manifolds) Sphere theorem (3-manifolds) Sphere theorem (3-manifolds) Sphere theorem (3-manifolds) Sphere theorem (3-manifolds) Sphere theorem (3-manifolds)
dbp:wikiPageUsesTemplate	dbt:About dbt:Cite_book dbt:Cite_journal dbt:Harv dbt:Short_description dbt:Harvs
authorlink	Christos Papakyriakopoulos (en)
first	Christos (en)
last	Papakyriakopoulos (en)
year	1957 (xsd:integer)
has abstract	In der Topologie, einem Teilgebiet der Mathematik, ist der Sphärensatz ein grundlegender Lehrsatz aus der Theorie 3-dimensionaler Mannigfaltigkeiten. Er wurde 1957 von Christos Papakyriakopoulos bewiesen. Ebenso wie der unter dem Namen Dehns Lemma bekannte Schleifensatz stellt er einen Zusammenhang zwischen der (in algebraischen Begriffen formulierbaren) Homotopietheorie und der geometrischen Topologie von 3-Mannigfaltigkeiten her; beide Sätze bilden die Grundlage für große Teile der Theorie der 3-Mannigfaltigkeiten. (de) In mathematics, in the topology of 3-manifolds, the sphere theorem of Christos Papakyriakopoulos 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 Jean-Loïc Batude. 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 1. * , 2. * , 3. * is a covering map, and 4. * is a 2-sided submanifold (2-sphere or projective plane) of . quoted in . (en) Теорема о сфере — классическое утверждение трёхмерной топологии, доказанное Христосом Папакирьякопулосом в 1956 году вместе с леммой Дена и теоремой о петле. (ru)
prov:wasDerivedFrom	wikipedia-en:Sphere_theorem_(3-manifolds)?oldid=1021893719&ns=0
page length (characters) of wiki page	3464 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Sphere_theorem_(3-manifolds)
is Link from a Wikipage to another Wikipage of	Dehn's lemma 3-manifold Christos Papakyriakopoulos Aspherical space Loop theorem
is foaf:primaryTopic of	wikipedia-en:Sphere_theorem_(3-manifolds)

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