View source for Kervaire invariant

An invariant of an almost-parallelizable smooth manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k0553501.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k0553502.png" />, defined as the [[Arf-invariant|Arf-invariant]] of the quadratic form modulo 2 on the lattice of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k0553503.png" />-dimensional homology space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k0553504.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k0553505.png" /> be a simply-connected almost-parallelizable closed smooth manifold of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k0553506.png" /> whose homology groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k0553507.png" /> vanish for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k0553508.png" />, except for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k0553509.png" />.

On the free Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535010.png" /> there is a skew-symmetric intersection form of cycles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535012.png" />, and the dimension of the integral lattice in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535013.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535014.png" />. There exists on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535015.png" /> a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535016.png" /> defined as follows: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535017.png" />, then there exists a smooth imbedding of the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535018.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535019.png" /> that realizes the given element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535021.png" />. A tubular neighbourhood of this sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535022.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535023.png" /> is parallelizable, and it can be either trivial or isomorphic to a tubular neighbourhood of the diagonal in the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535024.png" />. Here, the tubular neighbourhood of the diagonal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535025.png" /> is non-trivial if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535026.png" /> (see [[Hopf invariant|Hopf invariant]]). The value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535027.png" /> is zero or one depending on whether or not the tubular neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535028.png" /> realizing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535029.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535030.png" /> is trivial, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535031.png" />. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535032.png" /> satisfies the condition

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535033.png" /></td> </tr></table>

The Arf-invariant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535034.png" /> is also called the Kervaire invariant of the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535036.png" />.

If the Kervaire invariant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535037.png" /> is equal to zero, then there exists a symplectic basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535038.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535039.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535040.png" />. In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535041.png" /> is a connected sum of a product of spheres

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535042.png" /></td> </tr></table>

If, on the other hand, the Kervaire invariant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535043.png" /> is non-zero, then there is a symplectic basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535044.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535045.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535046.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535047.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535048.png" />. In this case the union of the tubular neighbourhoods of the two <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535049.png" />-dimensional spheres, imbedded in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535050.png" /> with transversal intersection at a point and realizing the elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535051.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535052.png" />, gives a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535053.png" />. It is called the Kervaire manifold (see [[Dendritic manifold|Dendritic manifold]]); its boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535054.png" /> is diffeomorphic to the standard sphere, while the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535055.png" /> itself can be expressed as the connected sum

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535056.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535057.png" /></td> </tr></table>

where the smooth closed manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535058.png" /> is obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535059.png" /> by attaching a cell.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535060.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535061.png" />, is a smooth parallelizable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535062.png" />-connected manifold with a boundary that is homotopic to a sphere, then the Kervaire invariant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535063.png" /> is defined exactly as above and will have the same properties with the difference that, in the decomposition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535064.png" /> into a connected sum of simple manifolds, the component <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535065.png" /> that is the Kervaire manifold has boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535066.png" /> (which generally is not diffeomorphic to the standard sphere).

In the cases <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535067.png" /> the original manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535068.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535069.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535070.png" /> can be expressed as the connected sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535071.png" /> (if the boundary is empty) or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535072.png" /> (if the boundary is non-empty), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535073.png" /> is obtained by removing an open cell from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535074.png" />.

However, a Kervaire invariant can be defined for the closed manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535075.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535076.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535077.png" /> (see [[Pontryagin invariant|Pontryagin invariant]]; [[Kervaire–Milnor invariant|Kervaire–Milnor invariant]]) and depends in these dimensions on the choice of the framing, that is, it is an invariant of the framed surgery of the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535078.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535079.png" />. In dimensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535080.png" /> the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535081.png" /> can be modified to the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535082.png" /> if and only if the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535083.png" /> has a framed surgery to the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535084.png" /> under any choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535085.png" /> on the original manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535086.png" /> (see [[Surgery|Surgery]] on a manifold).

The Kervaire invariant is defined for any stably-parallelizable manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535087.png" /> as an invariant of framed surgery, and any element in the stable homotopy groups of spheres can be represented either as a framed homotopy sphere or as a closed smooth framed Kervaire manifold (in this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535088.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535089.png" />), or as the framed manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535090.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535091.png" />.

In other words, the Kervaire invariant can be regarded as an obstruction to  "carrying over"  the given framing on the manifold to the sphere of the same dimension, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535092.png" />. In this sense the Kervaire invariant fulfills the same role for the values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535093.png" />: The given framing on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535094.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535095.png" />, cannot, in general, be  "carried over"  to the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535096.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535097.png" />, by means of framed surgery.

L.S. Pontryagin was the first to construct such a framing on the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535098.png" /> for the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k05535099.png" />, that is, a framing on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k055350100.png" />-dimensional torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k055350101.png" /> that cannot be  "carried over"  to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k055350102.png" />. There are also such examples of a framing on the manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k055350103.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k055350104.png" />.

The fundamental problem concerning the Kervaire invariant is the following: For which odd values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k055350105.png" /> does there exist a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k055350106.png" /> with non-zero Kervaire invariant? For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k055350107.png" /> the answer to this question is negative and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k055350108.png" /> it is affirmative, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k055350109.png" /> (Pontryagin, see [[#References|[2]]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k055350110.png" /> (M.A. Kervaire and J.W. Milnor, [[#References|[5]]], [[#References|[6]]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k055350111.png" /> (W. Browder, [[#References|[3]]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k055350112.png" /> (M. Barratt, M. Mahowald, A. Milgram). For other values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055350/k055350113.png" /> the answer is unknown (1989).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.P. Novikov,   "Homotopy-equivalent smooth manifolds I"  ''Izv. Akad. Nauk SSSR. Ser. Mat.'' , '''28''' :  2  (1964)  pp. 365–474  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.S. Pontryagin,   "Smooth manifolds and their applications in homology theory" , Moscow  (1976)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  W. Browder,   "The Kervaire invariant of framed manifolds and its generalization"  ''Ann. of Math.'' , '''90'''  (1969)  pp. 157–186</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  W.B. Browder,   "Surgery on simply-connected manifolds" , Springer  (1972)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  M. Kervaire,   "A manifold which does not admit any differentiable structure"  ''Comm. Math. Helv.'' , '''34'''  (1960)  pp. 257–270</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  M.A. Kervaire,   J.W. Milnor,   "Groups of homotopy spheres I"  ''Ann. Mat.'' , '''77''' :  3  (1963)  pp. 504–537</TD></TR></table>