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In mathematics, a 5-manifold is a 5-dimensional topological manifold, possibly with a piecewise linear or smooth structure. Non-simply connected 5-manifolds are impossible to classify, as this is harder than solving the word problem for groups. Simply connected compact 5-manifolds were first classified by Stephen Smale and then in full generality by Dennis Barden, while another proof was later given by Aleksey V. Zhubr. This turns out to be easier than the 3- or 4-dimensional case: the 3-dimensional case is the Thurston geometrisation conjecture, and the 4-dimensional case was solved by Michael Freedman (1982) in the topological case, but is a very hard unsolved problem in the smooth case.

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rdf:type	yago:WikicatManifolds yago:Artifact100021939 yago:Conduit103089014 yago:Manifold103717750 yago:Object100002684 yago:Passage103895293 yago:PhysicalEntity100001930 yago:Pipe103944672 yago:YagoGeoEntity yago:YagoPermanentlyLocatedEntity yago:Tube104493505 yago:Way104564698 yago:Whole100003553
rdfs:label	5-manifold (en)
rdfs:comment	In mathematics, a 5-manifold is a 5-dimensional topological manifold, possibly with a piecewise linear or smooth structure. Non-simply connected 5-manifolds are impossible to classify, as this is harder than solving the word problem for groups. Simply connected compact 5-manifolds were first classified by Stephen Smale and then in full generality by Dennis Barden, while another proof was later given by Aleksey V. Zhubr. This turns out to be easier than the 3- or 4-dimensional case: the 3-dimensional case is the Thurston geometrisation conjecture, and the 4-dimensional case was solved by Michael Freedman (1982) in the topological case, but is a very hard unsolved problem in the smooth case. (en)
dcterms:subject	Manifolds Geometric topology
Wikipage page ID	7763362 (xsd:integer)
Wikipage revision ID	1106806147 (xsd:integer)
Link from a Wikipage to another Wikipage	Dennis Barden Algebraic topology Homogeneous space Stiefel–Whitney class Mathematics Stephen Smale 4-manifold Manifolds Differential structure Geometric topology Word problem for groups Rotation group SO(3) Special unitary group Michael Freedman Topological manifold Undecidable problem Simply connected Piecewise linear structure 5-dimensional Thurston geometrisation conjecture
Link from a Wikipage to an external page	http://www.map.mpim-bonn.mpg.de/5-manifolds:_1-connected%7Ctitle=5-manifolds: http://www.map.mpim-bonn.mpg.de/Linking_form%7Ctitle=Linking
sameAs	5-manifold 5-manifold 5-manifold 5-manifold
dbp:wikiPageUsesTemplate	dbt:Cite_web dbt:Reflist dbt:Short_description dbt:Topology-stub
has abstract	In mathematics, a 5-manifold is a 5-dimensional topological manifold, possibly with a piecewise linear or smooth structure. Non-simply connected 5-manifolds are impossible to classify, as this is harder than solving the word problem for groups. Simply connected compact 5-manifolds were first classified by Stephen Smale and then in full generality by Dennis Barden, while another proof was later given by Aleksey V. Zhubr. This turns out to be easier than the 3- or 4-dimensional case: the 3-dimensional case is the Thurston geometrisation conjecture, and the 4-dimensional case was solved by Michael Freedman (1982) in the topological case, but is a very hard unsolved problem in the smooth case. In dimension 5, the smooth classification of simply connected manifolds is governed by classical algebraic topology. Namely, two simply connected, smooth 5-manifolds are diffeomorphic if and only if there exists an isomorphism of their second homology groups with integer coefficients, preserving the linking form and the second Stiefel–Whitney class. Moreover, any such isomorphism in second homology is induced by some diffeomorphism. It is undecidable if a given 5-manifold is homeomorphic to , the 5-sphere. (en)
prov:wasDerivedFrom	wikipedia-en:5-manifold?oldid=1106806147&ns=0
page length (characters) of wiki page	3702 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:5-manifold
is Link from a Wikipage to another Wikipage of	Dennis Barden Timeline of manifolds Stephen Smale 4-manifold Smoothness Michael Freedman List of undecidable problems Five-dimensional space
is foaf:primaryTopic of	wikipedia-en:5-manifold

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