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Difference between revisions of "Kervaire invariant"

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m (fixing superscripts)
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whose homology groups  $  H _ {i} ( M ;  \mathbf Z ) $
 
whose homology groups  $  H _ {i} ( M ;  \mathbf Z ) $
 
vanish for  $  0 < i < 4 k + 2 $,  
 
vanish for  $  0 < i < 4 k + 2 $,  
except for  $  V = H _ {2k+} 1 ( M ;  \mathbf Z ) $.
+
except for  $  V = H _ {2k+ 1} ( M ;  \mathbf Z ) $.
  
 
On the free Abelian group  $  V $
 
On the free Abelian group  $  V $
Line 30: Line 30:
 
a function  $  \Phi _ {0} :  V \rightarrow \mathbf Z _ {2} $
 
a function  $  \Phi _ {0} :  V \rightarrow \mathbf Z _ {2} $
 
defined as follows: If  $  x \in V $,  
 
defined as follows: If  $  x \in V $,  
then there exists a smooth imbedding of the sphere  $  S  ^ {2k+} 1 $
+
then there exists a smooth imbedding of the sphere  $  S  ^ {2k+ 1} $
 
into  $  M $
 
into  $  M $
 
that realizes the given element  $  x $,  
 
that realizes the given element  $  x $,  
 
$  k \geq  1 $.  
 
$  k \geq  1 $.  
A tubular neighbourhood of this sphere  $  S  ^ {2k+} 1 $
+
A tubular neighbourhood of this sphere  $  S  ^ {2k+ 1} $
 
in  $  M $
 
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 $.  
+
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 $
+
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 $(
 
is non-trivial if and only if  $  2 k + 1 \neq 1 , 3 , 7 $(
 
see [[Hopf invariant|Hopf invariant]]). The value of  $  \Phi _ {0} $
 
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 $
+
is zero or one depending on whether or not the tubular neighbourhood of  $  S  ^ {2k+ 1} $
 
realizing  $  x $
 
realizing  $  x $
 
in  $  M $
 
in  $  M $
Line 52: Line 52:
  
 
The Arf-invariant of  $  \Phi _ {0} $
 
The Arf-invariant of  $  \Phi _ {0} $
is also called the Kervaire invariant of the manifold  $  M  ^ {4k+} 2 $,  
+
is also called the Kervaire invariant of the manifold  $  M  ^ {4k+ 2} $,  
 
$  2 k + 1 \neq 1 , 3 , 7 $.
 
$  2 k + 1 \neq 1 , 3 , 7 $.
  
If the Kervaire invariant of  $  M  ^ {4k+} 2 $
+
If the Kervaire invariant of  $  M  ^ {4k+ 2} $
 
is equal to zero, then there exists a symplectic basis  $  ( e _ {i} , f _ {i} ) $
 
is equal to zero, then there exists a symplectic basis  $  ( e _ {i} , f _ {i} ) $
 
for  $  V $
 
for  $  V $
 
such that  $  \Phi _ {0} ( e _ {i} ) = \Phi _ {0} ( f _ {i} ) = 0 $.  
 
such that  $  \Phi _ {0} ( e _ {i} ) = \Phi _ {0} ( f _ {i} ) = 0 $.  
In this case  $  M  ^ {4k+} 2 $
+
In this case  $  M  ^ {4k+ 2} $
 
is a connected sum of a product of spheres
 
is a connected sum of a product of spheres
  
 
$$  
 
$$  
M  ^ {4k+} 2 = \  
+
M  ^ {4k+ 2}  = \  
( S  ^ {2k+} 1 \times S  ^ {2k+} 1 ) _ {1} \# \dots \#
+
( S  ^ {2k+ 1} \times S  ^ {2k+ 1} ) _ {1} \# \dots \#
( S  ^ {2k+} 1 \times S  ^ {2k+} 1 ) _ {m} .
+
( S  ^ {2k+ 1} \times S  ^ {2k+ 1} ) _ {m} .
 
$$
 
$$
  
If, on the other hand, the Kervaire invariant of  $  M  ^ {4k+} 2 $
+
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} ) $
 
is non-zero, then there is a symplectic basis  $  ( e _ {i} , f _ {i} ) $
 
for  $  V $
 
for  $  V $
Line 75: Line 75:
 
and  $  \Phi _ {0} ( e _ {1} ) = \Phi _ {0} ( f _ {1} ) = 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 ) $-
 
In this case the union of the tubular neighbourhoods of the two  $  ( 2 k + 1 ) $-
dimensional spheres, imbedded in  $  M  ^ {4k+} 2 $
+
dimensional spheres, imbedded in  $  M  ^ {4k+ 2} $
 
with transversal intersection at a point and realizing the elements  $  e _ {1} $,  
 
with transversal intersection at a point and realizing the elements  $  e _ {1} $,  
 
$  f _ {1} $,  
 
$  f _ {1} $,  
gives a manifold  $  K  ^ {4k+} 2 $.  
+
gives a manifold  $  K  ^ {4k+ 2} $.  
It is called the Kervaire manifold (see [[Dendritic manifold|Dendritic manifold]]); its boundary  $  \partial  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 $
+
is diffeomorphic to the standard sphere, while the manifold  $  M  ^ {4k+ 2} $
 
itself can be expressed as the connected sum
 
itself can be expressed as the connected sum
  
 
$$  
 
$$  
M  ^ {4k+} 2 = \  
+
M  ^ {4k+ 2}  = \  
\widehat{K}  {}  ^ {4k+} 2 \# ( S  ^ {2k+} 1 \times S  ^ {2k+} 1 ) _ {1} \# \dots
+
\widehat{K}  {}  ^ {4k+ 2} \# ( S  ^ {2k+ 1} \times S  ^ {2k+ 1} ) _ {1} \# \dots
 
$$
 
$$
  
 
$$  
 
$$  
{} \dots \# ( S  ^ {2k+} 1 \times S  ^ {2k+} 1 ) _ {m-} 1 ,
+
{} \dots \# ( S  ^ {2k+ 1} \times S  ^ {2k+ 1} ) _ {m- 1} ,
 
$$
 
$$
  
where the smooth closed manifold  $  \widehat{K}  {}  ^ {4k+} 2 $
+
where the smooth closed manifold  $  \widehat{K}  {}  ^ {4k+ 2} $
is obtained from  $  K  ^ {4k+} 2 $
+
is obtained from  $  K  ^ {4k+ 2} $
 
by attaching a cell.
 
by attaching a cell.
  
If  $  M  ^ {4k+} 2 $,  
+
If  $  M  ^ {4k+ 2} $,  
 
$  k \neq 0 , 1 , 3 $,  
 
$  k \neq 0 , 1 , 3 $,  
 
is a smooth parallelizable  $  ( 2 k ) $-
 
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 $
+
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 $
+
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 $
+
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 $(
+
that is the Kervaire manifold has boundary  $  \partial  K  ^ {4k+ 2} = \partial  M  ^ {4k+ 2} $(
 
which generally is not diffeomorphic to the standard sphere).
 
which generally is not diffeomorphic to the standard sphere).
  
Line 109: Line 109:
 
$  M  ^ {6} $,  
 
$  M  ^ {6} $,  
 
$  M  ^ {14} $
 
$  M  ^ {14} $
can be expressed as the connected sum  $  ( S  ^ {2k+} 1 \times S  ^ {2k+} 1 ) \# \dots \# ( S  ^ {2k+} 1 \times S  ^ {2k+} 1 ) $(
+
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 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} $
+
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 $.
+
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} $,  
 
However, a Kervaire invariant can be defined for the closed manifolds  $  M  ^ {2} $,  
 
$  M  ^ {6} $,  
 
$  M  ^ {6} $,  
 
$  M  ^ {14} $(
 
$  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} ) $,  
+
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 $.  
 
$  k = 0 , 1 , 3 $.  
 
In dimensions  $  k \neq 0 , 1 , 3 $
 
In dimensions  $  k \neq 0 , 1 , 3 $
the manifold  $  M  ^ {4k+} 2 $
+
the manifold  $  M  ^ {4k+ 2} $
can be modified to the sphere  $  S  ^ {4k+} 2 $
+
can be modified to the sphere  $  S  ^ {4k+ 2} $
if and only if the pair  $  ( M  ^ {4k+} 2 , f _ {r} ) $
+
if and only if the pair  $  ( M  ^ {4k+ 2} , f _ {r} ) $
has a framed surgery to the pair  $  ( S  ^ {4k+} 2 , f _ {r} ) $
+
has a framed surgery to the pair  $  ( S  ^ {4k+ 2} , f _ {r} ) $
 
under any choice of  $  f _ {r} $
 
under any choice of  $  f _ {r} $
on the original manifold  $  M  ^ {4k+} 2 $(
+
on the original manifold  $  M  ^ {4k+ 2} $(
 
see [[Surgery|Surgery]] on a manifold).
 
see [[Surgery|Surgery]] on a manifold).
  
The Kervaire invariant is defined for any stably-parallelizable manifold  $  M  ^ {4k+} 2 $
+
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 $,  
 
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 $),  
 
$  k \neq 0 , 1 , 3 $),  
or as the framed manifold  $  S  ^ {2k+} 1 \times S  ^ {2k+} 1 $
+
or as the framed manifold  $  S  ^ {2k+ 1} \times S  ^ {2k+ 1} $
 
if  $  k = 0 , 1 , 3 $.
 
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 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 $:  
 
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 $,  
+
The given framing on  $  S  ^ {2k+ 1} \times S  ^ {2k+ 1} $,  
 
$  k = 0 , 1 , 3 $,  
 
$  k = 0 , 1 , 3 $,  
cannot, in general, be  "carried over"  to the sphere  $  S  ^ {4k+} 2 $,  
+
cannot, in general, be  "carried over"  to the sphere  $  S  ^ {4k+ 2} $,  
 
$  k = 0 , 1 , 3 $,  
 
$  k = 0 , 1 , 3 $,  
 
by means of framed surgery.
 
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 $
+
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 $,  
 
for the case  $  k = 0 $,  
 
that is, a framing on the  $  2 $-
 
that is, a framing on the  $  2 $-
Line 154: Line 154:
 
with non-zero Kervaire invariant? For  $  n \neq 2  ^ {i} - 1 $
 
with non-zero Kervaire invariant? For  $  n \neq 2  ^ {i} - 1 $
 
the answer to this question is negative and for  $  n = 2  ^ {i} - 1 $
 
the answer to this question is negative and for  $  n = 2  ^ {i} - 1 $
it is affirmative, where  $  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).
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>

Revision as of 05:26, 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=51840
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