About: Kervaire invariant

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In mathematics, the Kervaire invariant is an invariant of a framed -dimensional manifold that measures whether the manifold could be surgically converted into a sphere. This invariant evaluates to 0 if the manifold can be converted to a sphere, and 1 otherwise. This invariant was named after Michel Kervaire who built on work of Cahit Arf. In any given dimension, there are only two possibilities: either all manifolds have Arf–Kervaire invariant equal to 0, or half have Arf–Kervaire invariant 0 and the other half have Arf–Kervaire invariant 1.

Attributes	Values
rdfs:label	Kervaire invariant (en)
rdfs:comment	In mathematics, the Kervaire invariant is an invariant of a framed -dimensional manifold that measures whether the manifold could be surgically converted into a sphere. This invariant evaluates to 0 if the manifold can be converted to a sphere, and 1 otherwise. This invariant was named after Michel Kervaire who built on work of Cahit Arf. In any given dimension, there are only two possibilities: either all manifolds have Arf–Kervaire invariant equal to 0, or half have Arf–Kervaire invariant 0 and the other half have Arf–Kervaire invariant 1. (en)
dcterms:subject	Surgery theory Differential topology
Wikipage page ID	13816037 (xsd:integer)
Wikipage revision ID	1081333272 (xsd:integer)
Link from a Wikipage to another Wikipage	Cahit Arf Scientific American Bernoulli number Annals of Mathematics Arf invariant Hopf fibration De Rham invariant J-homomorphism L-theory Isotropy index Quadratic refinement Quateroctonionic projective plane Surgery theory Notices of the American Mathematical Society Smooth structure Ergebnisse der Mathematik und ihrer Grenzgebiete Lev Pontryagin Commentarii Mathematici Helvetici Hopf invariant Kervaire manifold Torus H-cobordism Differential topology Exotic sphere Quadratic form Stable homotopy theory Surgery theory Homotopy group Homotopy sphere Topology (journal) Differentiable manifold Manifold Rosenfeld projective plane Michael Atiyah Michel Kervaire Signature (topology) Journal of the London Mathematical Society Immersion (mathematics) Symplectic basis Parallelizable manifold Steenrod square PL manifold Cayley projective plane Skew-quadratic form Stable homotopy group of spheres Homology group Ε-symmetric form dbr:Progress_in_Mathematics
Link from a Wikipage to an external page	http://www.scientificamerican.com/article.cfm%3Fid=hypersphere-exotica http://www-math.mit.edu/~hrm/ksem.html https://web.archive.org/web/20090511131847/http:/www.math.rochester.edu/u/faculty/doug/kervaire.html https://web.archive.org/web/20101221083231/http:/manifoldatlas.uni-bonn.de/Exotic_spheres http://golem.ph.utexas.edu/category/2009/04/kervaire_invariant_one_problem.html http://www.maths.ed.ac.uk/~aar/atiyah80.htm https://www.simonsfoundation.org/news/-/asset_publisher/bo1E/content/mathematicians-solve-45-year-old-kervaire-invariant-puzzle http://www.uni-math.gwdg.de/schick/publ/Groups%20of%20homotopy%20spheres%20I.pdf https://www.ams.org/notices/201106/rtx110600804p.pdf
sameAs	Kervaire invariant Kervaire invariant Kervaire invariant
dbp:wikiPageUsesTemplate	dbt:Springer dbt:As_of dbt:Citation dbt:Cite_journal dbt:Harvtxt dbt:Refbegin dbt:Refend dbt:Reflist dbt:Harvs dbt:Eom
authorlink	William Browder (en)
first	Michael J. (en) William (en) Michael A. (en) A.V. (en) M.A. (en) Douglas C. (en)
id	A/a013230 (en) K/k055350 (en) k/k055360 (en)
last	Hill (en) Hopkins (en) Ravenel (en) Browder (en) Chernavskii (en) Shtan'ko (en)
title	Arf invariant (en) Kervaire invariant (en) Kervaire-Milnor invariant (en)
year	1969 (xsd:integer) 2016 (xsd:integer)
has abstract	In mathematics, the Kervaire invariant is an invariant of a framed -dimensional manifold that measures whether the manifold could be surgically converted into a sphere. This invariant evaluates to 0 if the manifold can be converted to a sphere, and 1 otherwise. This invariant was named after Michel Kervaire who built on work of Cahit Arf. The Kervaire invariant is defined as the Arf invariant of the skew-quadratic form on the middle dimensional homology group. It can be thought of as the simply-connected quadratic L-group , and thus analogous to the other invariants from L-theory: the signature, a -dimensional invariant (either symmetric or quadratic, ), and the De Rham invariant, a -dimensional symmetric invariant . In any given dimension, there are only two possibilities: either all manifolds have Arf–Kervaire invariant equal to 0, or half have Arf–Kervaire invariant 0 and the other half have Arf–Kervaire invariant 1. The Kervaire invariant problem is the problem of determining in which dimensions the Kervaire invariant can be nonzero. For differentiable manifolds, this can happen in dimensions 2, 6, 14, 30, 62, and possibly 126, and in no other dimensions. The final case of dimension 126 remains open. (en)
author2-link	Michael J. Hopkins (en)
author3-link	Douglas Ravenel (en)

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