J-homomorphism
In mathematics, the J-homomorphism is a mapping from the homotopy groups of the special orthogonal groups to the homotopy [groups of spheres]. It was defined by, extending a construction of.
Definition
Whitehead's original homomorphism is defined geometrically, and gives a homomorphismof abelian groups for integers q, and.
The J-homomorphism can be defined as follows.
An element of the special orthogonal group SO can be regarded as a map
and the homotopy group ) consists of homotopy classes of maps from the r-sphere to SO.
Thus an element of can be represented by a map
Applying the Hopf construction to this gives a map
in, which Whitehead defined as the image of the element of under the J-homomorphism.
Taking a limit as q tends to infinity gives the stable J-homomorphism in stable [homotopy theory]:
where is the infinite special orthogonal group, and the right-hand side is the r-th stable stem of the stable [homotopy groups of spheres].
Image of the J-homomorphism
The image of the J-homomorphism was described by, assuming the Adams conjecture of which was proved by, as follows. The group is given by Bott periodicity. It is always cyclic; and if r is positive, it is of order 2 if r is 0 or 1 modulo 8, infinite if r is 3 or 7 modulo 8, and order 1 otherwise. In particular the image of the stable J-homomorphism is cyclic. The stable homotopy groups are the direct sum of the image of the J-homomorphism, and the kernel of the Adams e-invariant, a homomorphism from the stable homotopy groups to. If r is 0 or 1 mod 8 and positive, the order of the image is 2. If r is 3 or 7 mod 8, the image is a cyclic group of order equal to the denominator of, where is a Bernoulli number. In the remaining cases where r is 2, 4, 5, or 6 mod 8 the image is trivial because is trivial.Applications
introduced the group J of a space X, which for X a sphere is the image of the J-homomorphism in a suitable dimension.The cokernel of the J-homomorphism appears in the group Θn of h-cobordism classes of oriented homotopy n-spheres.