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 homomorphism
of 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 SO 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 mod 8, infinite if r is 3 mod 4, 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. The order of the image is 2 if r is 0 or 1 mod 8 and positive. If is 3 mod 4 and positive 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 of exotic spheres.