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 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 .)

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 ) consists of homotopy classes of maps from the r-sphere to SO(q). 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 Frank Adams (1966), assuming the Adams conjecture of Adams (1963) which was proved by Daniel Quillen (1971), 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 (Switzer 1975, p. 488). In particular the image of the stable J-homomorphism is cyclic. The stable homotopy groups πrS 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 Q/Z. 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 r = 4n−1 is 3 mod 4 and positive the image is a cyclic group of order equal to the denominator of B2n/4n, where B2n is a Bernoulli number. In the remaining cases where r is 2, 4, 5, or 6 mod 8 the image is trivial because πr(SO) is trivial.

r 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
πr(SO) 121Z111Z221Z111Z22
|im(J)| 1212411124022150411148022
πrS Z2224112240222365041322480×22224
B2n 16130142130


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.

The cokernel of the J-homomorphism appears in the group of exotic spheres (Kosinski (1992)).


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