Kervaire invariant

An invariant of an almost-parallelizable smooth manifold of dimension , defined as the Arf-invariant of the quadratic form modulo 2 on the lattice of the -dimensional homology space of .

Let be a simply-connected almost-parallelizable closed smooth manifold of dimension whose homology groups vanish for , except for .

On the free Abelian group there is a skew-symmetric intersection form of cycles , , and the dimension of the integral lattice in is equal to . There exists on a function defined as follows: If , then there exists a smooth imbedding of the sphere into that realizes the given element , . A tubular neighbourhood of this sphere in is parallelizable, and it can be either trivial or isomorphic to a tubular neighbourhood of the diagonal in the product . Here, the tubular neighbourhood of the diagonal in is non-trivial if and only if (see Hopf invariant). The value of is zero or one depending on whether or not the tubular neighbourhood of realizing in is trivial, . The function satisfies the condition

The Arf-invariant of is also called the Kervaire invariant of the manifold , .

If the Kervaire invariant of is equal to zero, then there exists a symplectic basis for such that . In this case is a connected sum of a product of spheres

If, on the other hand, the Kervaire invariant of is non-zero, then there is a symplectic basis for such that for and . In this case the union of the tubular neighbourhoods of the two -dimensional spheres, imbedded in with transversal intersection at a point and realizing the elements , , gives a manifold . It is called the Kervaire manifold (see Dendritic manifold); its boundary is diffeomorphic to the standard sphere, while the manifold itself can be expressed as the connected sum

where the smooth closed manifold is obtained from by attaching a cell.

If , , is a smooth parallelizable -connected manifold with a boundary that is homotopic to a sphere, then the Kervaire invariant of is defined exactly as above and will have the same properties with the difference that, in the decomposition of into a connected sum of simple manifolds, the component that is the Kervaire manifold has boundary (which generally is not diffeomorphic to the standard sphere).

In the cases the original manifolds , , can be expressed as the connected sum (if the boundary is empty) or (if the boundary is non-empty), where is obtained by removing an open cell from .

However, a Kervaire invariant can be defined for the closed manifolds , , (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 , . In dimensions the manifold can be modified to the sphere if and only if the pair has a framed surgery to the pair under any choice of on the original manifold (see Surgery on a manifold).

The Kervaire invariant is defined for any stably-parallelizable manifold 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 , ), or as the framed manifold if .

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, . In this sense the Kervaire invariant fulfills the same role for the values : The given framing on , , cannot, in general, be "carried over" to the sphere , , by means of framed surgery.

L.S. Pontryagin was the first to construct such a framing on the manifold for the case , that is, a framing on the -dimensional torus that cannot be "carried over" to . There are also such examples of a framing on the manifolds and .

The fundamental problem concerning the Kervaire invariant is the following: For which odd values of does there exist a pair with non-zero Kervaire invariant? For the answer to this question is negative and for it is affirmative, where (Pontryagin, see [2]), (M.A. Kervaire and J.W. Milnor, [5], [6]), (W. Browder, [3]), (M. Barratt, M. Mahowald, A. Milgram). For other values of 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