Kervaire invariant

Kervaire invariant

In mathematics, the Kervaire invariant, named for Michel Kervaire, is defined in geometric topology. It is an invariant of 4"n" + 2 dimensional almost-parallelizable smooth manifolds "M", taking values in the 2-element group

:Z/2Z.

It is equal to the Arf invariant of the quadratic form on the homology group

:"H"2"n"+1("M").

W. harvtxt|Browder|1969 proved that the Kervaire invariant vanishes unless "n" + 2 is a power of 2. It is nonzero for some manifold of dimension 2"k" − 2 for

:"k" = 1, 2, 3, 4, 5, 6 harv|Barratt|Jones|Mahowald|1984;

the question of in which dimensions there are manifolds of non-zero Kervaire invariant is called the Kervaire invariant problem. harvtxt|Kervaire|1960 used his invariant in 10 dimensions to find the first example of a PL manifold with no smooth structure.

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/2Z, and a homomorphism from the 14th stable homotopy group of spheres onto Z/2Z. For "n" = 2, 6, 14 there is anexotic framing on "S"n/2 x "S"n/2 with Kervaire-Milnor invariant 1.

References

*citation|id=MR|0810962
last=Barratt|first= M. G.|last2= Jones|first2= J. D. S.|last3= Mahowald|first3= M. E.
title=Relations amongst Toda brackets and the Kervaire invariant in dimension 62|journal=J. London Math. Soc. (2) |volume=30 |year=1984|issue= 3|pages= 533-550.

*citation|first=W. B.|last=Browder|title=The Kervaire invariant of framed manifolds and its generalization|journal= Ann. of Math. |volume= 90 |year=1969|pages= 157–186
url=http://links.jstor.org/sici?sici=0003-486X%28196907%292%3A90%3A1%3C157%3ATKIOFM%3E2.0.CO%3B2-W

*citation|first=W.B. |last=Browder|title=Surgery on simply-connected manifolds|publisher= Springer |year=1972|id=MR|0358813
series=Ergebnisse der Mathematik und ihrer Grenzgebiete|volume=65|publication-place= New York-Heidelberg|pp=ix+132|ISBN= 978-0387056296

* citation|first=M. |last=Kervaire|title=A manifold which does not admit any differentiable structure|journal= Comm. Math. Helv. |volume=34 |year=1960|pages= 257–270
url=http://retro.seals.ch/digbib/view?did=c1:391766&sdid=c1:392119

*springer|title=Kervaire invariant|id=K/k055350|first=M.A. |last=Shtan'ko
*springer|title=Kervaire-Milnor invariant|id=k/k055360|first=M.A. |last=Shtan'ko


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