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 r \ge 2. (Hopf defined this for the special case q = r+1.) 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 \mathrm{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 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 \pi_r^S 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 \Q/\Z. 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 B_{2n}/4n, where B_{2n} 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. ! style="text-align:right;width:10%" | r ! style="width:5%" | 0 ! style="width:5%" | 1 ! style="width:5%" | 2 ! style="width:5%" | 3 ! style="width:5%" | 4 ! style="width:5%" | 5 ! style="width:5%" | 6 ! style="width:5%" | 7 ! style="width:5%" | 8 ! style="width:5%" | 9 ! style="width:5%" | 10 ! style="width:5%" | 11 ! style="width:5%" | 12 ! style="width:5%" | 13 ! style="width:5%" | 14 ! style="width:5%" | 15 ! style="width:5%" | 16 ! style="width:5%" | 17 ! style="text-align:right" | ! style="text-align:right" | ! style="text-align:right" | \pi_r^S ! style="text-align:right" | B_{2n}

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 Θn of h-cobordism classes of oriented homotopy n-spheres.