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An invariant of framed [[Surgery|surgery]] of a closed 6- or 14 $-
+
An invariant of framed [[Surgery|surgery]] of a closed 6- or 14-dimensional framed manifold.
dimensional framed manifold.
 
  
 
Let  $  M  ^ {6} $
 
Let  $  M  ^ {6} $
be a stably-parallelizable $ 2 $-
+
be a stably-parallelizable  2-connected manifold on which is given a stable  $  N $-dimensional framing  $  ( M  ^ {6} , U ) $,  
connected manifold on which is given a stable  $  N $-
+
i.e. a trivialization of the stable  $  N $-dimensional [[Normal bundle|normal bundle]]. Let  $  S _ {i}  ^ {3} $
dimensional framing  $  ( M  ^ {6} , U ) $,  
+
be spheres realizing a basis of the  3-dimensional homology space of  $  M  ^ {6} $.  
i.e. a trivialization of the stable  $  N $-
+
By summing the given  $  N $-trivialization  $  U $
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} ) $
 
with certain trivializations  $  \alpha _ {i} \in \pi _ {3} (  \mathop{\rm SO} _ {3} ) $
 
of tubular neighbourhoods of the spheres  $  S _ {i}  ^ {3} $
 
of tubular neighbourhoods of the spheres  $  S _ {i}  ^ {3} $
 
in  $  M  ^ {6} $,  
 
in  $  M  ^ {6} $,  
one obtains an  $  ( N + 3 ) $-
+
one obtains an  $  ( N + 3 ) $-dimensional trivialization of the stable normal bundles to the spheres  $  S _ {i}  ^ {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} ) $.  
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 ) $
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} $
 
is isomorphic to  $  \mathbf Z _ {2} $
 
for  $  n = 3 $,  
 
for  $  n = 3 $,  
 
so that each sphere  $  S _ {i}  ^ {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 $(
+
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} $
according to the value of the elements  $  \alpha _ {i}  ^ {1} $
 
 
which they take in the group  $  \mathbf Z _ {2} $
 
which they take in the group  $  \mathbf Z _ {2} $
 
after factorization by  $  \alpha _ {i}  ^ {1} $).  
 
after factorization by  $  \alpha _ {i}  ^ {1} $).  
Line 45: Line 37:
 
$  \mathop{\rm mod}  2 $,  
 
$  \mathop{\rm mod}  2 $,  
 
where  $  \phi ( x , y ) $
 
where  $  \phi ( x , y ) $
is the intersection form of the 3 $-
+
is the intersection form of the 3-dimensional homology space on the manifold  $  M  ^ {6} $,  
dimensional homology space on the manifold  $  M  ^ {6} $,  
 
 
and is called the Kervaire–Milnor invariant of this manifold with framing  $  U $.  
 
and is called the Kervaire–Milnor invariant of this manifold with framing  $  U $.  
 
The pair  $  ( M  ^ {6} , U ) $
 
The pair  $  ( M  ^ {6} , U ) $
Line 54: Line 45:
  
 
Similar constructions have been carried out for  $  M  ^ {14} $.  
 
Similar constructions have been carried out for  $  M  ^ {14} $.  
The Kervaire–Milnor invariant in dimension six is the only invariant of the stable 6 $-
+
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} $,  
dimensional framed cobordism and defines an isomorphism  $  \pi _ {n+} 6 ( S  ^ {n} ) \approx \mathbf Z _ {2} $,  
 
 
$  n \geq  7 $.  
 
$  n \geq  7 $.  
However, in dimension fourteen it is not a unique invariant of the stable 14 $-
+
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} ) $,  
dimensional framed cobordism, i.e. the stable group  $  \pi _ {n+} 14 ( S  ^ {n} ) $,  
 
 
$  n \geq  16 $,  
 
$  n \geq  16 $,  
 
is defined by framings on the sphere  $  S  ^ {14} $
 
is defined by framings on the sphere  $  S  ^ {14} $

Revision as of 17:52, 20 January 2022


An invariant of framed 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. 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 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.

How to Cite This Entry:
Kervaire-Milnor invariant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kervaire-Milnor_invariant&oldid=51936
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