# 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 $\mathrm {SO}$ 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 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 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 $\pi _{r}^{S}$ 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 $\mathbb {Q} /\mathbb {Z}$. 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 $\pi _{r}(\operatorname {SO} )$ is trivial.
    r 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 $\pi _{r}(\operatorname {SO} )$ 121$\mathbb {Z}$111$\mathbb {Z}$221$\mathbb {Z}$111$\mathbb {Z}$22 $|\operatorname {im} (J)|$ 1212411124022150411148022 $\pi _{r}^{S}$ $\mathbb {Z}$22241122402$^2$2$^3$6504132$^2$480×22$^2$2$^4$ $B_{2n}$ $^1$⁄$_6$−$^1$⁄$_{30}$$^1$⁄$_{42}$−$^1$⁄$_{30}$

## Applications

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.
The cokernel of the J-homomorphism $J\colon \pi _{n}(\mathrm {SO} )\to \pi _{n}^{S}$ appears in the group Θ$_n$ of h-cobordism classes of oriented homotopy n-spheres (Kosinski (1992)).