View source for Kervaire-Milnor invariant

An invariant of framed [[Surgery|surgery]] of a closed 6- or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055360/k0553601.png" />-dimensional framed manifold.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055360/k0553602.png" /> be a stably-parallelizable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055360/k0553603.png" />-connected manifold on which is given a stable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055360/k0553605.png" />-dimensional framing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055360/k0553606.png" />, i.e. a trivialization of the stable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055360/k0553607.png" />-dimensional [[Normal bundle|normal bundle]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055360/k0553608.png" /> be spheres realizing a basis of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055360/k0553609.png" />-dimensional homology space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055360/k05536010.png" />. By summing the given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055360/k05536011.png" />-trivialization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055360/k05536012.png" /> with certain trivializations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055360/k05536013.png" /> of tubular neighbourhoods of the spheres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055360/k05536014.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055360/k05536015.png" />, one obtains an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055360/k05536016.png" />-dimensional trivialization of the stable normal bundles to the spheres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055360/k05536017.png" /> and the corresponding elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055360/k05536018.png" />. The cokernel of the stable homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055360/k05536019.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055360/k05536020.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055360/k05536021.png" />, so that each sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055360/k05536022.png" /> is put into correspondence with an element of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055360/k05536023.png" /> (according to the value of the elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055360/k05536024.png" /> which they take in the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055360/k05536025.png" /> after factorization by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055360/k05536026.png" />). This value does not depend on the choice of the elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055360/k05536027.png" />, but depends only on the homology classes realized by the spheres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055360/k05536028.png" /> and the framing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055360/k05536029.png" />. The [[Arf-invariant|Arf-invariant]] of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055360/k05536030.png" /> so obtained satisfies the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055360/k05536031.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055360/k05536032.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055360/k05536033.png" /> is the intersection form of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055360/k05536034.png" />-dimensional homology space on the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055360/k05536035.png" />, and is called the Kervaire–Milnor invariant of this manifold with framing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055360/k05536036.png" />. The pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055360/k05536037.png" /> has a framed surgery to the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055360/k05536038.png" /> if and only if the Kervaire–Milnor invariant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055360/k05536039.png" /> is zero.

Similar constructions have been carried out for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055360/k05536040.png" />. The Kervaire–Milnor invariant in dimension six is the only invariant of the stable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055360/k05536041.png" />-dimensional framed cobordism and defines an isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055360/k05536042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055360/k05536043.png" />. However, in dimension fourteen it is not a unique invariant of the stable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055360/k05536044.png" />-dimensional framed cobordism, i.e. the stable group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055360/k05536045.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055360/k05536046.png" />, is defined by framings on the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055360/k05536047.png" /> and on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055360/k05536048.png" />.

For references see [[Kervaire invariant|Kervaire invariant]].