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 formula_2. (Hopf defined this for the special case formula_3.)
The "J"-homomorphism can be defined as follows. 
An element of the special orthogonal group SO("q") can be regarded as a map
and the homotopy group formula_5) consists of homotopy classes of maps from the "r"-sphere to SO("q").
Thus an element of formula_5 can be represented by a map
Applying the Hopf construction to this gives a map
in formula_9, which Whitehead defined as the image of the element of formula_5 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 formula_12 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 formula_13 are the direct sum of the (cyclic) image of the "J"-homomorphism, and the kernel of the Adams e-invariant , a homomorphism from the stable homotopy groups to formula_14. The order of the image is 2 if "r" is 0 or 1 mod 8 and positive (so in this case the "J"-homomorphism is injective). If formula_15 is 3 mod 4 and positive the image is a cyclic group of order equal to the denominator of formula_16, where formula_17 is a Bernoulli number. In the remaining cases where "r" is 2, 4, 5, or 6 mod 8 the image is trivial because formula_12 is trivial.
Applications.
 introduced the group "J"("X") 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 ().