It was defined by George W. Whitehead (1942), extending a construction of Heinz Hopf (1935).
Whitehead's original homomorphism is defined geometrically, and gives a homomorphism of abelian groups for integers q, and
(Hopf defined this for the special case
An element of the special orthogonal group SO(q) can be regarded as a map and the homotopy group
) consists of homotopy classes of maps from the r-sphere to SO(q).
, which Whitehead defined as the image of the element of
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.
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 (Switzer 1975, p. 488).
In particular the image of the stable J-homomorphism is cyclic.
The stable homotopy groups
are the direct sum of the (cyclic) image of the J-homomorphism, and the kernel of the Adams e-invariant (Adams 1966), 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 (so in this case the J-homomorphism is injective).
If r is 3 or 7 mod 8, the image is a cyclic group of order equal to the denominator of
In the remaining cases where r is 2, 4, 5, or 6 mod 8 the image is trivial because
Michael Atiyah (1961) 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.
appears in the group Θn of h-cobordism classes of oriented homotopy n-spheres (Kosinski (1992)).