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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" />.
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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" />.
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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
+
An invariant of an almost-parallelizable smooth manifold  $  M $
 +
of dimension  $  k + 2 $,  
 +
defined as the [[Arf-invariant|Arf-invariant]] of the quadratic form modulo 2 on the lattice of the  $  ( 2 k + 1 ) $-dimensional homology space of  $  M $.
  
<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>
+
Let  $  M $
 +
be a simply-connected almost-parallelizable closed smooth manifold of dimension  $  4 k + 2 $
 +
whose homology groups  $  H _ {i} ( M ; \mathbf Z ) $
 +
vanish for  $  0 < i < 4 k + 2 $,
 +
except for  $  V = H _ {2k+ 1} ( M ; \mathbf Z ) $.
  
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" />.
+
On the free Abelian group  $  V $
 +
there is a skew-symmetric intersection form of cycles  $  \Phi ( x , y ) $,
 +
$  \Phi :  V \times V \rightarrow \mathbf Z $,
 +
and the dimension of the integral lattice in  $  V $
 +
is equal to  $  2 m $.
 +
There exists on  $  V $
 +
a function  $  \Phi _ {0} : V \rightarrow \mathbf Z _ {2} $
 +
defined as follows: If  $  x \in V $,
 +
then there exists a smooth imbedding of the sphere  $  S  ^ {2k+ 1} $
 +
into  $  M $
 +
that realizes the given element  $  x $,
 +
$  k \geq  1 $.  
 +
A tubular neighbourhood of this sphere  $  S  ^ {2k+ 1} $
 +
in  $  M $
 +
is parallelizable, and it can be either trivial or isomorphic to a tubular neighbourhood of the diagonal in the product  $  S  ^ {2k+ 1} \times S  ^ {2k+ 1} $.  
 +
Here, the tubular neighbourhood of the diagonal in  $  S  ^ {2k+ 1} \times S  ^ {2k+ 1} $
 +
is non-trivial if and only if  $  2 k + 1 \neq 1 , 3 , 7 $ (see [[Hopf invariant|Hopf invariant]]). The value of  $  \Phi _ {0} $
 +
is zero or one depending on whether or not the tubular neighbourhood of $  S  ^ {2k+ 1} $
 +
realizing  $  x $
 +
in  $  M $
 +
is trivial,  $  2 k + 1 \neq 1 , 3 , 7 $.  
 +
The function  $  \Phi _ {0} : V \rightarrow \mathbf Z _ {2} $
 +
satisfies the condition
  
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
+
$$
 +
\Phi _ {0} ( x + y )  \equiv  \Phi _ {0} ( x) + \Phi _ {0} ( y) + \Phi ( x , y )  \mathop{\rm mod}  2 .
 +
$$
  
<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>
+
The Arf-invariant of  $  \Phi _ {0} $
 +
is also called the Kervaire invariant of the manifold  $  M  ^ {4k+ 2} $,
 +
$  2 k + 1 \neq 1 , 3 , 7 $.
  
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
+
If the Kervaire invariant of $  M  ^ {4k+ 2} $
 +
is equal to zero, then there exists a symplectic basis $  ( e _ {i} , f _ {i} ) $
 +
for $  V $
 +
such that $  \Phi _ {0} ( e _ {i} ) = \Phi _ {0} ( f _ {i} ) = 0 $.  
 +
In this case $  M  ^ {4k+ 2} $
 +
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/k05535056.png" /></td> </tr></table>
+
$$
 +
M  ^ {4k+ 2}  = \
 +
( S  ^ {2k+ 1} \times S  ^ {2k+ 1} ) _ {1} \# \dots \#
 +
( S  ^ {2k+ 1} \times S  ^ {2k+ 1} ) _ {m} .
 +
$$
  
<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>
+
If, on the other hand, the Kervaire invariant of  $  M  ^ {4k+ 2} $
 +
is non-zero, then there is a symplectic basis  $  ( e _ {i} , f _ {i} ) $
 +
for  $  V $
 +
such that  $  \Phi _ {0} ( e _ {i} ) = \Phi _ {0} ( f _ {i} ) = 0 $
 +
for  $  i \neq 1 $
 +
and  $  \Phi _ {0} ( e _ {1} ) = \Phi _ {0} ( f _ {1} ) = 1 $.  
 +
In this case the union of the tubular neighbourhoods of the two  $  ( 2 k + 1 ) $-
 +
dimensional spheres, imbedded in  $  M  ^ {4k+ 2} $
 +
with transversal intersection at a point and realizing the elements  $  e _ {1} $,
 +
$  f _ {1} $,
 +
gives a manifold  $  K  ^ {4k+ 2} $.  
 +
It is called the Kervaire manifold (see [[Dendritic manifold|Dendritic manifold]]); its boundary  $  \partial  K  ^ {4k+ 2} $
 +
is diffeomorphic to the standard sphere, while the manifold  $  M  ^ {4k+ 2} $
 +
itself can be expressed as the connected sum
  
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.
+
$$
 +
M  ^ {4k+ 2}  = \
 +
\widehat{K}  {}  ^ {4k+ 2} \# ( S  ^ {2k+ 1} \times S  ^ {2k+ 1} ) _ {1} \# \dots
 +
$$
  
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).
+
$$
 +
{} \dots \# ( S  ^ {2k+ 1} \times S  ^ {2k+ 1} ) _ {m- 1} ,
 +
$$
  
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" />.
+
where the smooth closed manifold  $  \widehat{K}  {}  ^ {4k+ 2} $
 +
is obtained from  $  K  ^ {4k+ 2} $
 +
by attaching a cell.
  
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).
+
If  $  M  ^ {4k+ 2} $,  
 +
$  k \neq 0 , 1 , 3 $,  
 +
is a smooth parallelizable  $  ( 2 k ) $-connected manifold with a boundary that is homotopic to a sphere, then the Kervaire invariant of $  M  ^ {4k+ 2} $
 +
is defined exactly as above and will have the same properties with the difference that, in the decomposition of  $  M  ^ {4k+ 2} $
 +
into a connected sum of simple manifolds, the component  $  K _ {0}  ^ {4k+ 2} $
 +
that is the Kervaire manifold has boundary  $  \partial  K  ^ {4k+ 2} = \partial  M  ^ {4k+ 2} $ (which generally is not diffeomorphic to the standard sphere).
  
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 the cases  $  k = 0 , 1 , 3 $
 +
the original manifolds  $  M  ^ {2} $,
 +
$  M  ^ {6} $,
 +
$  M  ^ {14} $
 +
can be expressed as the connected sum  $  ( S  ^ {2k+ 1} \times S  ^ {2k+ 1} ) \# \dots \# ( S  ^ {2k+ 1} \times S  ^ {2k+ 1} ) $(
 +
if the boundary is empty) or ( S  ^ {2k+ 1} \times S  ^ {2k+ 1} ) _ {0} \# \dots \# ( S  ^ {2k+ 1} \times S  ^ {2k+ 1} ) _ {m- 1} $(
 +
if the boundary is non-empty), where  $  ( S  ^ {2k+ 1} \times S  ^ {2k+ 1} ) _ {0} $
 +
is obtained by removing an open cell from  $  S  ^ {2k+ 1} \times S  ^ {2k+ 1} $.
  
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.
+
However, a Kervaire invariant can be defined for the closed manifolds  $  M  ^ {2} $,
 +
$  M  ^ {6} $,
 +
$ M ^ {14} $ (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  $  ( M  ^ {4k+ 2} , f _ {r} ) $,
 +
k = 0 , 1 , 3 $.  
 +
In dimensions  $  k \neq 0 , 1 , 3 $
 +
the manifold  $ M ^ {4k+ 2} $
 +
can be modified to the sphere $  S  ^ {4k+ 2} $
 +
if and only if the pair  $  ( M  ^ {4k+ 2} , f _ {r} ) $
 +
has a framed surgery to the pair  $  ( S  ^ {4k+ 2} , f _ {r} ) $
 +
under any choice of $  f _ {r} $
 +
on the original manifold  $  M  ^ {4k+ 2} $ (see [[Surgery|Surgery]] on a manifold).
  
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 Kervaire invariant is defined for any stably-parallelizable manifold  $  M  ^ {4k+ 2} $
 +
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  $  m = 4 k + 2 $,
 +
k \neq 0 , 1 , 3 $),  
 +
or as the framed manifold  $  S  ^ {2k+ 1} \times S  ^ {2k+ 1} $
 +
if $ k = 0 , 1 , 3 $.
  
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).
+
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,  $  k \neq 0 , 1 , 3 $.
 +
In this sense the Kervaire invariant fulfills the same role for the values  $  k = 0 , 1 , 3 $:
 +
The given framing on  $  S  ^ {2k+ 1} \times S  ^ {2k+ 1} $,
 +
$  k = 0 , 1 , 3 $,
 +
cannot, in general, be  "carried over"  to the sphere  $  S  ^ {4k+ 2} $,
 +
$  k = 0 , 1 , 3 $,
 +
by means of framed surgery.
 +
 
 +
L.S. Pontryagin was the first to construct such a framing on the manifold  $  S  ^ {2k+ 1} \times S  ^ {2k+ 1} $
 +
for the case  $  k = 0 $,
 +
that is, a framing on the  $  2 $-
 +
dimensional torus  $  ( ( S  ^ {1} \times S  ^ {1} ) , f _ {r} ) $
 +
that cannot be  "carried over"  to  $  S  ^ {2} $.
 +
There are also such examples of a framing on the manifolds  $  S  ^ {3} \times S  ^ {3} $
 +
and  $  S  ^ {7} \times S  ^ {7} $.
 +
 
 +
The fundamental problem concerning the Kervaire invariant is the following: For which odd values of $  n $
 +
does there exist a pair $  ( M  ^ {2n} , f _ {r} ) $
 +
with non-zero Kervaire invariant? For $  n \neq 2  ^ {i} - 1 $
 +
the answer to this question is negative and for $  n = 2  ^ {i} - 1 $
 +
it is affirmative, where $  i = 1 $ (Pontryagin, see [[#References|[2]]]), $  i = 2 , 3 $ (M.A. Kervaire and J.W. Milnor, [[#References|[5]]], [[#References|[6]]]), $  i = 4 $ (W. Browder, [[#References|[3]]]), $  i = 5 , 6 $ (M. Barratt, M. Mahowald, A. Milgram). For other values of $  i $ the answer is unknown (1989).
  
 
====References====
 
====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>
 
<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>

Latest revision as of 05:30, 4 January 2022


An invariant of an almost-parallelizable smooth manifold $ M $ of dimension $ k + 2 $, defined as the Arf-invariant of the quadratic form modulo 2 on the lattice of the $ ( 2 k + 1 ) $-dimensional homology space of $ M $.

Let $ M $ be a simply-connected almost-parallelizable closed smooth manifold of dimension $ 4 k + 2 $ whose homology groups $ H _ {i} ( M ; \mathbf Z ) $ vanish for $ 0 < i < 4 k + 2 $, except for $ V = H _ {2k+ 1} ( M ; \mathbf Z ) $.

On the free Abelian group $ V $ there is a skew-symmetric intersection form of cycles $ \Phi ( x , y ) $, $ \Phi : V \times V \rightarrow \mathbf Z $, and the dimension of the integral lattice in $ V $ is equal to $ 2 m $. There exists on $ V $ a function $ \Phi _ {0} : V \rightarrow \mathbf Z _ {2} $ defined as follows: If $ x \in V $, then there exists a smooth imbedding of the sphere $ S ^ {2k+ 1} $ into $ M $ that realizes the given element $ x $, $ k \geq 1 $. A tubular neighbourhood of this sphere $ S ^ {2k+ 1} $ in $ M $ is parallelizable, and it can be either trivial or isomorphic to a tubular neighbourhood of the diagonal in the product $ S ^ {2k+ 1} \times S ^ {2k+ 1} $. Here, the tubular neighbourhood of the diagonal in $ S ^ {2k+ 1} \times S ^ {2k+ 1} $ is non-trivial if and only if $ 2 k + 1 \neq 1 , 3 , 7 $ (see Hopf invariant). The value of $ \Phi _ {0} $ is zero or one depending on whether or not the tubular neighbourhood of $ S ^ {2k+ 1} $ realizing $ x $ in $ M $ is trivial, $ 2 k + 1 \neq 1 , 3 , 7 $. The function $ \Phi _ {0} : V \rightarrow \mathbf Z _ {2} $ satisfies the condition

$$ \Phi _ {0} ( x + y ) \equiv \Phi _ {0} ( x) + \Phi _ {0} ( y) + \Phi ( x , y ) \mathop{\rm mod} 2 . $$

The Arf-invariant of $ \Phi _ {0} $ is also called the Kervaire invariant of the manifold $ M ^ {4k+ 2} $, $ 2 k + 1 \neq 1 , 3 , 7 $.

If the Kervaire invariant of $ M ^ {4k+ 2} $ is equal to zero, then there exists a symplectic basis $ ( e _ {i} , f _ {i} ) $ for $ V $ such that $ \Phi _ {0} ( e _ {i} ) = \Phi _ {0} ( f _ {i} ) = 0 $. In this case $ M ^ {4k+ 2} $ is a connected sum of a product of spheres

$$ M ^ {4k+ 2} = \ ( S ^ {2k+ 1} \times S ^ {2k+ 1} ) _ {1} \# \dots \# ( S ^ {2k+ 1} \times S ^ {2k+ 1} ) _ {m} . $$

If, on the other hand, the Kervaire invariant of $ M ^ {4k+ 2} $ is non-zero, then there is a symplectic basis $ ( e _ {i} , f _ {i} ) $ for $ V $ such that $ \Phi _ {0} ( e _ {i} ) = \Phi _ {0} ( f _ {i} ) = 0 $ for $ i \neq 1 $ and $ \Phi _ {0} ( e _ {1} ) = \Phi _ {0} ( f _ {1} ) = 1 $. In this case the union of the tubular neighbourhoods of the two $ ( 2 k + 1 ) $- dimensional spheres, imbedded in $ M ^ {4k+ 2} $ with transversal intersection at a point and realizing the elements $ e _ {1} $, $ f _ {1} $, gives a manifold $ K ^ {4k+ 2} $. It is called the Kervaire manifold (see Dendritic manifold); its boundary $ \partial K ^ {4k+ 2} $ is diffeomorphic to the standard sphere, while the manifold $ M ^ {4k+ 2} $ itself can be expressed as the connected sum

$$ M ^ {4k+ 2} = \ \widehat{K} {} ^ {4k+ 2} \# ( S ^ {2k+ 1} \times S ^ {2k+ 1} ) _ {1} \# \dots $$

$$ {} \dots \# ( S ^ {2k+ 1} \times S ^ {2k+ 1} ) _ {m- 1} , $$

where the smooth closed manifold $ \widehat{K} {} ^ {4k+ 2} $ is obtained from $ K ^ {4k+ 2} $ by attaching a cell.

If $ M ^ {4k+ 2} $, $ k \neq 0 , 1 , 3 $, is a smooth parallelizable $ ( 2 k ) $-connected manifold with a boundary that is homotopic to a sphere, then the Kervaire invariant of $ M ^ {4k+ 2} $ is defined exactly as above and will have the same properties with the difference that, in the decomposition of $ M ^ {4k+ 2} $ into a connected sum of simple manifolds, the component $ K _ {0} ^ {4k+ 2} $ that is the Kervaire manifold has boundary $ \partial K ^ {4k+ 2} = \partial M ^ {4k+ 2} $ (which generally is not diffeomorphic to the standard sphere).

In the cases $ k = 0 , 1 , 3 $ the original manifolds $ M ^ {2} $, $ M ^ {6} $, $ M ^ {14} $ can be expressed as the connected sum $ ( S ^ {2k+ 1} \times S ^ {2k+ 1} ) \# \dots \# ( S ^ {2k+ 1} \times S ^ {2k+ 1} ) $( if the boundary is empty) or $ ( S ^ {2k+ 1} \times S ^ {2k+ 1} ) _ {0} \# \dots \# ( S ^ {2k+ 1} \times S ^ {2k+ 1} ) _ {m- 1} $( if the boundary is non-empty), where $ ( S ^ {2k+ 1} \times S ^ {2k+ 1} ) _ {0} $ is obtained by removing an open cell from $ S ^ {2k+ 1} \times S ^ {2k+ 1} $.

However, a Kervaire invariant can be defined for the closed manifolds $ M ^ {2} $, $ M ^ {6} $, $ M ^ {14} $ (see Pontryagin 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 $ ( M ^ {4k+ 2} , f _ {r} ) $, $ k = 0 , 1 , 3 $. In dimensions $ k \neq 0 , 1 , 3 $ the manifold $ M ^ {4k+ 2} $ can be modified to the sphere $ S ^ {4k+ 2} $ if and only if the pair $ ( M ^ {4k+ 2} , f _ {r} ) $ has a framed surgery to the pair $ ( S ^ {4k+ 2} , f _ {r} ) $ under any choice of $ f _ {r} $ on the original manifold $ M ^ {4k+ 2} $ (see Surgery on a manifold).

The Kervaire invariant is defined for any stably-parallelizable manifold $ M ^ {4k+ 2} $ 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 $ m = 4 k + 2 $, $ k \neq 0 , 1 , 3 $), or as the framed manifold $ S ^ {2k+ 1} \times S ^ {2k+ 1} $ if $ k = 0 , 1 , 3 $.

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, $ k \neq 0 , 1 , 3 $. In this sense the Kervaire invariant fulfills the same role for the values $ k = 0 , 1 , 3 $: The given framing on $ S ^ {2k+ 1} \times S ^ {2k+ 1} $, $ k = 0 , 1 , 3 $, cannot, in general, be "carried over" to the sphere $ S ^ {4k+ 2} $, $ k = 0 , 1 , 3 $, by means of framed surgery.

L.S. Pontryagin was the first to construct such a framing on the manifold $ S ^ {2k+ 1} \times S ^ {2k+ 1} $ for the case $ k = 0 $, that is, a framing on the $ 2 $- dimensional torus $ ( ( S ^ {1} \times S ^ {1} ) , f _ {r} ) $ that cannot be "carried over" to $ S ^ {2} $. There are also such examples of a framing on the manifolds $ S ^ {3} \times S ^ {3} $ and $ S ^ {7} \times S ^ {7} $.

The fundamental problem concerning the Kervaire invariant is the following: For which odd values of $ n $ does there exist a pair $ ( M ^ {2n} , f _ {r} ) $ with non-zero Kervaire invariant? For $ n \neq 2 ^ {i} - 1 $ the answer to this question is negative and for $ n = 2 ^ {i} - 1 $ it is affirmative, where $ i = 1 $ (Pontryagin, see [2]), $ i = 2 , 3 $ (M.A. Kervaire and J.W. Milnor, [5], [6]), $ i = 4 $ (W. Browder, [3]), $ i = 5 , 6 $ (M. Barratt, M. Mahowald, A. Milgram). For other values of $ i $ the answer is unknown (1989).

References

[1] S.P. Novikov, "Homotopy-equivalent smooth manifolds I" Izv. Akad. Nauk SSSR. Ser. Mat. , 28 : 2 (1964) pp. 365–474 (In Russian)
[2] L.S. Pontryagin, "Smooth manifolds and their applications in homology theory" , Moscow (1976) (In Russian)
[3] W. Browder, "The Kervaire invariant of framed manifolds and its generalization" Ann. of Math. , 90 (1969) pp. 157–186
[4] W.B. Browder, "Surgery on simply-connected manifolds" , Springer (1972)
[5] M. Kervaire, "A manifold which does not admit any differentiable structure" Comm. Math. Helv. , 34 (1960) pp. 257–270
[6] M.A. Kervaire, J.W. Milnor, "Groups of homotopy spheres I" Ann. Mat. , 77 : 3 (1963) pp. 504–537
How to Cite This Entry:
Kervaire invariant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kervaire_invariant&oldid=14005
This article was adapted from an original article by M.A. Shtan'ko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article