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, extending a construction of.

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 $$r \ge 2$$. (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
 * $$S^{q-1}\rightarrow S^{q-1}$$

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, assuming the Adams conjecture of which was proved by , 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. 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 $$\pi_r(\operatorname{SO})$$ is trivial.


 * {| class="wikitable" style="text-align: center; background-color:white"

! 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" | $$\pi_r(\operatorname{SO})$$ ! style="text-align:right" | $$|\operatorname{im}(J)|$$ ! style="text-align:right" | $$\pi_r^S$$ ! style="text-align:right" | $$B_{2n}$$
 * 1 || 2 || 1 || $$\Z$$ || 1 || 1 || 1 || $$\Z$$ || 2 || 2 || 1 || $$\Z$$ || 1 || 1 || 1 || $$\Z$$ || 2 || 2
 * 1 || 2 || 1 || 24 || 1 || 1 || 1 || 240 || 2 || 2 || 1 || 504 || 1 || 1 || 1 || 480 || 2 || 2
 * $$\Z$$ || 2 || 2 || 24 || 1 || 1 || 2 || 240 || 22 || 23 || 6 || 504 || 1 || 3 || 22 || 480×2 || 22 || 24
 * || ||  || 1⁄6 ||  ||  ||  || −1⁄30 ||  ||  ||  || 1⁄42 ||  ||  ||  || −1⁄30 ||  ||
 * }

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 $$J \colon \pi_n(\mathrm{SO}) \to \pi_n^S$$ appears in the group Θn of h-cobordism classes of oriented homotopy n-spheres.