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
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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
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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
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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 |
Kervaire invariant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kervaire_invariant&oldid=47495