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| In mathematics, the '''Kervaire invariant''', named for [[Michel Kervaire]], is defined in [[geometric topology]]. It is an invariant of a (4''k''+2)-dimensional ([[singly even]]-dimensional) [[framed manifold|framed]] differentiable manifold (or more generally PL-manifold) ''M,'' taking values in the 2-element group '''Z'''/2'''Z''' = {0,1}. 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-theory|L-group]] <math>L_{4k+2},</math> and thus analogous to the other invariants from L-theory: the [[signature (topology)|signature]], a 4''k''-dimensional invariant (either symmetric or quadratic, <math>L^{4k} \cong L_{4k}</math>), and the [[De Rham invariant]], a (4''k''+1)-dimensional ''symmetric'' invariant <math>L^{4k+1}.</math>
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| 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.
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| == Definition ==
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| The Kervaire invariant is the [[Arf invariant]] of the [[quadratic form]] determined by the framing on the middle-dimensional ''Z''/2''Z''-coefficient homology group
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| :''q'' : ''H''<sub>2''m''+1</sub>(''M'';''Z''/2''Z'') <math>\to</math> ''Z''/2''Z'',
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| and is thus sometimes called the '''Arf–Kervaire invariant'''. The quadratic form (properly, [[skew-quadratic form]]) is a [[quadratic refinement]] of the usual [[ε-symmetric form]] on the middle dimensional homology of an (unframed) even-dimensional manifold; the framing yields the quadratic refinement.
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| The quadratic form ''q'' can be defined by algebraic topology using functional [[Steenrod square]]s, and geometrically via the self-intersections
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| of [[Immersion (mathematics)|immersions]] <math>S^{2m+1}</math><math>\to</math> <math>M^{4m+2}</math> determined by the framing, or by the triviality/non-triviality of the normal bundles of embeddings <math>S^{2m+1}</math><math>\to</math> <math>M^{4m+2}</math> (for <math>m \neq 0,1,3</math>) and the mod 2 [[Hopf invariant]] of maps <math>S^{4m+2+k} \to S^{2m+1+k}</math>
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| (for <math>m = 0,1,3</math>).
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| == History ==
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| The Kervaire invariant is a generalization of the Arf invariant of a framed surface (= 2-dimensional manifold with stably trivialized tangent bundle) which was used by [[Pontryagin]] in 1950 to compute of the [[homotopy group]] <math>\pi_{n+2}(S^n)=Z/2Z</math> of maps <math>S^{n+2}</math> <math>\to</math> <math>S^n</math> (for <math>n\geq 2</math>), which is the cobordism group of surfaces embedded in <math>S^{n+2}</math> with trivialized normal bundle.
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| {{harvtxt|Kervaire|1960}} used his invariant for ''n'' = 10 to construct the [[Kervaire manifold]], a 10-dimensional [[PL manifold]] with no [[smooth structure|differentiable structure]], the first example of such a manifold, by showing that his invariant does not vanish on this PL manifold, but vanishes on all smooth manifolds of dimension 10.
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| {{harvtxt|Kervaire|Milnor|1963}} computes the group of [[exotic sphere]]s (in dimension greater than 4), with one step in the computation depending on the Kervaire invariant problem. Specifically, they show that the set of exotic spheres of dimension ''n'' – specifically the monoid of smooth structures on the standard ''n''-sphere – is isomorphic to the group Θ<sub>''n''</sub> of [[H-cobordism|''h''-cobordism]] classes of oriented [[homotopy sphere|homotopy ''n''-spheres]]. They compute this latter in terms of a map
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| :<math>\Theta_n/bP_{n+1}\to \pi_n^S/J,\, </math>
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| where <math>bP_{n+1}</math> is the cyclic subgroup of ''n''-spheres that bound a [[parallelizable manifold]] of dimension ''n''+1, <math>\pi_n^S</math> is the ''n''th [[stable homotopy group of spheres]], and ''J'' is the image of the [[J-homomorphism]], which is also a cyclic group. The <math>bP_{n+1}</math> and <math>J</math> are easily understood cyclic factors, which are trivial or order two except in dimension <math>n = 4k+3,</math> in which case they are large, with order related to [[Bernoulli number]]s. The quotients are the difficult parts of the groups. The map between these quotient groups is either an isomorphism or is injective and has an image of index 2. It is the latter if and only if there is an ''n''-dimensional framed manifold of nonzero Kervaire invariant, and thus the classification of exotic spheres depends up to a factor of 2 on the Kervaire invariant problem.
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| == Examples ==
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| For the standard embedded [[torus]], the skew-symmetric form is given by <math>\begin{pmatrix}0 & 1\\-1 & 0\end{pmatrix}</math> (with respect to the standard [[symplectic basis]]), and the skew-quadratic refinement is given by <math>xy</math> with respect to this basis: <math>Q(1,0)=Q(0,1)=0</math>: the basis curves don't self-link; and <math>Q(1,1)=1</math>: a (1,1) self-links, as in the [[Hopf fibration]]. This form thus has [[Arf invariant]] 0 (most of its elements have norm 0; it has [[isotropy index]] 1), and thus the standard embedded torus has Kervaire invariant 0.
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| == Kervaire invariant problem ==
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| The question of in which dimensions ''n'' there are ''n''-dimensional framed manifolds of nonzero Kervaire invariant is called the '''Kervaire invariant problem'''. This is only possible if ''n'' is 2 mod 4, and indeed one must have ''n'' is 2<sup>''k''</sup> − ''2'' (two less than a power of two). The question is almost completely resolved; {{as of|2012|lc=1}} only the case of dimension 126 is open: there are manifolds with nonzero Kervaire invariant in dimension 2, 6, 14, 30, 62, and none in all other dimensions other than possibly 126.
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| The main results are {{harvtxt|Browder|1969}}, which reduced the problem from differential topology to [[stable homotopy theory]] and showed that the only possible dimensions are 2<sup>''k''</sup> − ''2'', and {{harvtxt|Hill|Hopkins|Ravenel|2009}}, which showed that there were no such manifolds for <math>k \geq 8</math> (<math>n \geq 254</math>). Together with explicit constructions for lower dimensions (through 62), this leaves open only dimension 126.
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| It is conjectured by [[Michael Atiyah]] that there is such a manifold in dimension 126, and that the higher-dimensional manifolds with nonzero Kervaire invariant are related to well-known exotic manifolds two dimension higher, in dimensions 16, 32, 64, and 128, namely the [[Cayley projective plane]] <math>\mathbf{O}P^2</math> (dimension 16, octonionic projective plane) and the analogous [[Rosenfeld projective plane]]s (the bi-octonionic projective plane in dimension 32, the quater-octonionic projective plane in dimension 64, and the octo-octonionic projective plane in dimension 128), specifically that there is a construction that takes these projective planes and produces a manifold with nonzero Kervaire invariant in two dimensions lower.<ref>[http://mathoverflow.net/questions/101031/kervaire-invariant-why-dimension-126-especially-difficult#comment-259820 comment] by André Henriques Jul 1, 2012 at 19:26, on "[http://mathoverflow.net/questions/101031/kervaire-invariant-why-dimension-126-especially-difficult Kervaire invariant: Why dimension 126 especially difficult?]", ''[http://mathoverflow.net/ MathOverflow]''</ref>
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| === History ===
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| *{{harvtxt|Kervaire|1960}} proved that the Kervaire invariant is zero for manifolds of dimension 10, 18
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| *{{harvtxt|Kervaire|Milnor|1963}} proved that the Kervaire invariant can be nonzero for manifolds of dimension 6, 14
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| *{{harvtxt|Anderson|Brown|Peterson|1966}} proved that the Kervaire invariant is zero for manifolds of dimension 8''n''+2 for ''n''>1
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| *{{harvtxt|Mahowald|Tangora|1967}} proved that the Kervaire invariant can be nonzero for manifolds of dimension 30
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| *{{harvtxt|Browder|1969}} proved that the Kervaire invariant is zero for manifolds of dimension ''n'' not of the form 2<sup>''k''</sup> − ''2''.
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| *{{harvtxt|Barratt|Jones|Mahowald|1984}} showed that the Kervaire invariant is nonzero for some manifold of dimension 62.
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| *{{harvtxt|Hill|Hopkins|Ravenel|2009}} showed that the Kervaire invariant is zero for ''n''-dimensional framed manifolds for ''n'' = 2<sup>''k''</sup>− 2 with ''k'' ≥ 8. They constructed a cohomology theory Ω with the following properties from which their result follows immediately:
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| **The coefficient groups Ω<sup>''n''</sup>(point) have period 2<sup>8</sup>=256 in ''n''
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| **The coefficient groups Ω<sup>''n''</sup>(point) have a "gap": they vanish for ''n''=1, 2, 3
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| **The coefficient groups Ω<sup>''n''</sup>(point) can detect non-vanishing Kervaire invariants: more precisely if the Kervaire invariant for manifolds of dimension ''n'' is nonzero then it has a nonzero image in Ω<sup>−''n''</sup>(point)
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| == Kervaire–Milnor invariant ==
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| The '''Kervaire–Milnor''' invariant is a closely related invariant of framed surgery of a 2, 6 or 14-dimensional framed manifold, that gives isomorphisms from the 2nd and 6th [[stable homotopy group of spheres]] to '''Z'''/2'''Z''',
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| and a homomorphism from the 14th stable homotopy group of spheres onto '''Z'''/2'''Z'''. For ''n'' = 2, 6, 14 there is an
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| exotic framing on ''S''<sup>n/2</sup> x ''S''<sup>n/2</sup> with Kervaire-Milnor invariant 1.
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| ==See also==
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| * [[Signature (topology)|Signature]], a 4''k''-dimensional invariant
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| * [[De Rham invariant]], a (4''k''+1)-dimensional invariant
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| ==References==
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| {{reflist}}
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| {{refbegin}}
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| *{{cite doi|10.1112/jlms/s2-30.3.533}}
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| *{{cite jstor|1970686}}
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| *{{citation|first=W.|last=Browder|title=Surgery on simply-connected manifolds|publisher= Springer |year=1972|mr=0358813 |series=[[Ergebnisse der Mathematik und ihrer Grenzgebiete]]|volume=65|publication-place= New York-Heidelberg|pages=ix+132|isbn= 978-0-387-05629-6}}
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| *{{eom|id=A/a013230|title=Arf invariant|first=A.V. |last=Chernavskii}}
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| *{{cite arxiv| last1=Hill | first1=Michael A. | last2=Hopkins | first2=Michael J. |author2-link = Michael J. Hopkins | last3=Ravenel | first3=Douglas C. | title=On the non-existence of elements of Kervaire invariant one | eprint=0908.3724 | year=2009| class=math.AT | ref = harv }}
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| *{{cite doi|10.1007/BF02565940}}
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| *{{cite jstor|1970128}}
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| *{{cite doi|10.1016/0040-9383(67)90023-7}}
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| *{{Citation | last1=Miller | first1=Haynes | title=Kervaire Invariant One (after M. A. Hill, M. J. Hopkins, and D. C. Ravenel) | origyear=2011 | url=http://arxiv.org/abs/1104.4523 | series=Seminaire Bourbaki | year=2012}}
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| *{{citation|authorlink=John Milnor|first=John W.|last= Milnor |title=Differential topology forty-six years later|journal= [[Notices of the American Mathematical Society]] |volume=58|year=2011|issue= 6 |pages=804–809|url=http://www.ams.org/notices/201106/rtx110600804p.pdf}}
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| *Rourke, C. P. and [[Dennis Sullivan|Sullivan, D. P.]], On the Kervaire obstruction, Ann. of Math. (2) 94, 397—413 (1971)
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| *{{springer|title=Kervaire invariant|id=K/k055350|first=M.A. |last=Shtan'ko}}
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| *{{springer|title=Kervaire-Milnor invariant|id=k/k055360|first=M.A. |last=Shtan'ko}}
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| *{{Citation | last1=Snaith | first1=Victor P. | title=Stable homotopy around the Arf-Kervaire invariant | publisher=Birkhäuser Verlag | series=[[Progress in Mathematics]] | isbn=978-3-7643-9903-0 | mr=2498881 | year=2009 | volume=273}}
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| *{{Citation | last1=Snaith | first1=Victor P. | title=The Arf-Kervaire Invariant of framed manifolds | arxiv=1001.4751 | year=2010}}
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| {{refend}}
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| ==External links==
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| *[http://www.maths.ed.ac.uk/~aar/atiyah80.htm Slides and video of lecture by Hopkins at Edinburgh, 21 April, 2009]
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| *[http://www.math.rochester.edu/u/faculty/doug/kervaire.html Arf-Kervaire home page of Doug Ravenel]
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| *[http://www-math.mit.edu/~hrm/ksem.html Harvard-MIT Summer Seminar on the Kervaire Invariant]
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| *[http://golem.ph.utexas.edu/category/2009/04/kervaire_invariant_one_problem.html 'Kervaire Invariant One Problem' Solved], April 23, 2009, blog post by John Baez and discussion, The n-Category Café
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| *[http://manifoldatlas.uni-bonn.de/Exotic_spheres Exotic spheres] at the manifold atlas
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| ===Popular news stories===
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| * [http://www.scientificamerican.com/article.cfm?id=hypersphere-exotica Hypersphere Exotica: Kervaire Invariant Problem Has a Solution! A 45-year-old problem on higher-dimensional spheres is solved–probably], by Davide Castelvecchi, August 2009 ''[[Scientific American]]''
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| * {{cite doi|10.1038/news.2009.427}}
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| * [https://www.simonsfoundation.org/news/-/asset_publisher/bo1E/content/mathematicians-solve-45-year-old-kervaire-invariant-puzzle Mathematicians solve 45-year-old Kervaire invariant puzzle], Erica Klarreich, 20 Jul 2009
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| [[Category:Differential topology]]
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| [[Category:Surgery theory]]
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