# 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 George W. Whitehead (1942), extending a construction of Heinz Hopf (1935).

## Definition

Whitehead's original homomorphism is defined geometrically, and gives a homomorphism

$J\colon \pi _{r}(\mathrm {SO} (q))\to \pi _{r+q}(S^{q})\,\!$ of abelian groups for integers q, and $r\geq 2$ . (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

$S^{q-1}\rightarrow S^{q-1}$ and the homotopy group $\pi _{r}(\operatorname {SO} (q))$ ) consists of homotopy classes of maps from the r-sphere to SO(q). Thus an element of $\pi _{r}(\operatorname {SO} (q))$ can be represented by a map

$S^{r}\times S^{q-1}\rightarrow S^{q-1}$ Applying the Hopf construction to this gives a map

$S^{r+q}=S^{r}*S^{q-1}\rightarrow S(S^{q-1})=S^{q}$ in $\pi _{r+q}(S^{q})$ , which Whitehead defined as the image of the element of $\pi _{r}(\operatorname {SO} (q))$ under the J-homomorphism.

Taking a limit as q tends to infinity gives the stable J-homomorphism in stable homotopy theory:

$J\colon \pi _{r}(\mathrm {SO} )\to \pi _{r}^{S},\,\!$ 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 $\pi _{r}(\operatorname {SO} )$ 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

## Applications

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)).