Kervaire invariant

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