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 . (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 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 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 . 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 , 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.
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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 ().
References
.
External links
J-homomorphism at the nLab