About: Generalized Poincaré conjecture

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In the mathematical area of topology, the generalized Poincaré conjecture is a statement that a manifold which is a homotopy sphere is a sphere. More precisely, one fixes a category of manifolds: topological (Top), piecewise linear (PL), or differentiable (Diff). Then the statement is Every homotopy sphere (a closed n-manifold which is homotopy equivalent to the n-sphere) in the chosen category (i.e. topological manifolds, PL manifolds, or smooth manifolds) is isomorphic in the chosen category (i.e. homeomorphic, PL-isomorphic, or diffeomorphic) to the standard n-sphere.

Attributes	Values
rdf:type	yago:WikicatConjectures yago:Abstraction100002137 yago:Cognition100023271 yago:Concept105835747 yago:Content105809192 yago:Hypothesis105888929 yago:Idea105833840 yago:PsychologicalFeature100023100 yago:Speculation105891783 ArtificialSatellite
rdfs:label	Generalized Poincaré conjecture (en)
rdfs:comment	In the mathematical area of topology, the generalized Poincaré conjecture is a statement that a manifold which is a homotopy sphere is a sphere. More precisely, one fixes a category of manifolds: topological (Top), piecewise linear (PL), or differentiable (Diff). Then the statement is Every homotopy sphere (a closed n-manifold which is homotopy equivalent to the n-sphere) in the chosen category (i.e. topological manifolds, PL manifolds, or smooth manifolds) is isomorphic in the chosen category (i.e. homeomorphic, PL-isomorphic, or diffeomorphic) to the standard n-sphere. (en)
dcterms:subject	Conjectures Geometric topology Homotopy theory
Wikipage page ID	8536216 (xsd:integer)
Wikipage revision ID	1109287310 (xsd:integer)
Link from a Wikipage to another Wikipage	Millennium Prize Problems John R. Stallings Conjectures Mathematics Max Newman Christopher Zeeman Clay Mathematics Institute N-sphere Stephen Smale Closed manifold Topology Exotic sphere Fields Medal Fields medal Gluck twist Simply connected space Geometric topology Homotopy theory Grigori Perelman John Milnor Homotopy Homotopy sphere Differentiable manifold Manifold Piecewise linear manifold Poincaré conjecture Michael Freedman Michel Kervaire Orientability Category (mathematics) Topological manifold Differentiable structure Steve Smale
sameAs	Generalized Poincaré conjecture Generalized Poincaré conjecture Generalized Poincaré conjecture
dbp:wikiPageUsesTemplate	dbt:As_of dbt:Em dbt:Reflist dbt:Short_description
has abstract	In the mathematical area of topology, the generalized Poincaré conjecture is a statement that a manifold which is a homotopy sphere is a sphere. More precisely, one fixes a category of manifolds: topological (Top), piecewise linear (PL), or differentiable (Diff). Then the statement is Every homotopy sphere (a closed n-manifold which is homotopy equivalent to the n-sphere) in the chosen category (i.e. topological manifolds, PL manifolds, or smooth manifolds) is isomorphic in the chosen category (i.e. homeomorphic, PL-isomorphic, or diffeomorphic) to the standard n-sphere. The name derives from the Poincaré conjecture, which was made for (topological or PL) manifolds of dimension 3, where being a homotopy sphere is equivalent to being simply connected and closed. The generalized Poincaré conjecture is known to be true or false in a number of instances, due to the work of many distinguished topologists, including the Fields medal awardees John Milnor, Steve Smale, Michael Freedman, and Grigori Perelman. (en)
gold:hypernym	Sphere
prov:wasDerivedFrom	wikipedia-en:Generalized_Poincaré_conjecture?oldid=1109287310&ns=0
page length (characters) of wiki page	10547 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Generalized_Poincaré_conjecture
is Link from a Wikipage to another Wikipage of	John R. Stallings List of things named after Henri Poincaré Max Newman Geometric topology Conjecture Stephen Smale Mazur manifold H-cobordism 3-manifold Exotic R4 Fields Medal Differential structure Differential topology Grigori Perelman Homotopy sphere Manifold Piecewise linear manifold Poincaré conjecture Classification of manifolds Michael Freedman Generalized Poincare Conjecture Generalized Poincare conjecture Generalized Poincaré Conjecture Surgery exact sequence List of unsolved problems in mathematics Surgery structure set Smooth poincare conjecture
is Wikipage redirect of	Generalized Poincare Conjecture Generalized Poincare conjecture Generalized Poincaré Conjecture Smooth poincare conjecture
is known for of	Stephen Smale
is generalizations of	Poincaré conjecture
is known for of	John R. Stallings Stephen Smale Michael Freedman
is foaf:primaryTopic of	wikipedia-en:Generalized_Poincaré_conjecture

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