View source for Kervaire-Milnor invariant

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An invariant of framed [[Surgery|surgery]] of a closed 6- or 14-dimensional framed manifold.

Let  $  M  ^ {6} $
be a stably-parallelizable  2-connected manifold on which is given a stable  $  N $-dimensional framing  $  ( M  ^ {6} , U ) $, 
i.e. a trivialization of the stable  $  N $-dimensional [[Normal bundle|normal bundle]]. Let  $  S _ {i}  ^ {3} $
be spheres realizing a basis of the  3-dimensional homology space of  $  M  ^ {6} $. 
By summing the given  $  N $-trivialization  $  U $
with certain trivializations  $  \alpha _ {i} \in \pi _ {3} (  \mathop{\rm SO} _ {3} ) $
of tubular neighbourhoods of the spheres  $  S _ {i}  ^ {3} $
in  $  M  ^ {6} $, 
one obtains an  $  ( N + 3 ) $-dimensional trivialization of the stable normal bundles to the spheres  $  S _ {i}  ^ {3} $
and the corresponding elements  $  \alpha _ {i}  ^ {1} \in \pi _ {3} (  \mathop{\rm SO} _ {N+ 3} ) $. 
The cokernel of the stable homomorphism  $  s :  \pi _ {n} (  \mathop{\rm SO} _ {N+ 3} ) \rightarrow \pi _ {n} (  \mathop{\rm SO} _ {N+} n ) $
is isomorphic to  $  \mathbf Z _ {2} $
for  $  n = 3 $, 
so that each sphere  $  S _ {i}  ^ {3} $
is put into correspondence with an element of the group  $  \pi _ {3} (  \mathop{\rm SO} _ {N+ 3} ) /  \mathop{\rm Im}  s $ (according to the value of the elements  $  \alpha _ {i}  ^ {1} $
which they take in the group  $  \mathbf Z _ {2} $
after factorization by  $  \alpha _ {i}  ^ {1} $). 
This value does not depend on the choice of the elements  $  \alpha _ {i} $, 
but depends only on the homology classes realized by the spheres  $  S  ^ {3} $
and the framing  $  U $. 
The [[Arf-invariant|Arf-invariant]] of the function  $  \phi _ {0} :  H _ {3} ( M  ^ {6} , \mathbf Z ) \rightarrow \mathbf Z _ {2} $
so obtained satisfies the formula  $  \phi _ {0} ( x + y ) = \phi _ {0} ( x) + \phi _ {0} ( y) + \phi ( x, y ) $
$   \mathop{\rm mod}  2 $, 
where  $  \phi ( x , y ) $
is the intersection form of the 3-dimensional homology space on the manifold  $  M  ^ {6} $, 
and is called the Kervaire–Milnor invariant of this manifold with framing  $  U $. 
The pair  $  ( M  ^ {6} , U ) $
has a framed surgery to the pair  $  ( S  ^ {6} , V ) $
if and only if the Kervaire–Milnor invariant of  $  ( M  ^ {6} , U ) $
is zero.

Similar constructions have been carried out for  $  M  ^ {14} $. 
The Kervaire–Milnor invariant in dimension six is the only invariant of the stable 6-dimensional framed cobordism and defines an isomorphism  $  \pi _ {n+ 6} ( S  ^ {n} ) \approx \mathbf Z _ {2} $, 
$  n \geq  7 $. 
However, in dimension fourteen it is not a unique invariant of the stable 14-dimensional framed cobordism, i.e. the stable group  $  \pi _ {n+14} ( S  ^ {n} ) $, 
$  n \geq  16 $, 
is defined by framings on the sphere  $  S  ^ {14} $
and on  $  S  ^ {7} \times S  ^ {7} $.

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