# 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

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 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 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 π_{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 **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* = 4*n*−1 is 3 mod 4 and positive the image is a cyclic group of order equal to the denominator of *B*_{2n}/4*n*, 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 π_{r}(SO) is trivial.

r 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 π _{r}(SO)1 2 1 **Z**1 1 1 **Z**2 2 1 **Z**1 1 1 **Z**2 2 |im( *J*)|1 2 1 24 1 1 1 240 2 2 1 504 1 1 1 480 2 2 π _{r}^{S}**Z**2 2 24 1 1 2 240 2 ^{2}2 ^{3}6 504 1 3 2 ^{2}480×2 2 ^{2}2 ^{4}*B*_{2n}^{1}⁄_{6}− ^{1}⁄_{30}^{1}⁄_{42}− ^{1}⁄_{30}

## 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)).

## References

- Atiyah, Michael Francis (1961), "Thom complexes",
*Proceedings of the London Mathematical Society*, Third Series,**11**: 291–310, doi:10.1112/plms/s3-11.1.291, ISSN 0024-6115, MR 0131880 - Adams, J. F. (1963), "On the groups J(X) I",
*Topology*,**2**(3): 181, doi:10.1016/0040-9383(63)90001-6 - Adams, J. F. (1965a), "On the groups J(X) II",
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*Algebraic Topology—Homotopy and Homology*, Springer-Verlag, ISBN 978-0-387-06758-2 - Whitehead, George W. (1942), "On the homotopy groups of spheres and rotation groups",
*Annals of Mathematics*, Second Series,**43**(4): 634–640, doi:10.2307/1968956, ISSN 0003-486X, JSTOR 1968956, MR 0007107 - Whitehead, George W. (1978),
*Elements of homotopy theory*, Berlin: Springer, ISBN 0-387-90336-4, MR 0516508