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

${\displaystyle J\colon \pi _{r}(\mathrm {SO} (q))\to \pi _{r+q}(S^{q})}$

of abelian groups for integers q, and ${\displaystyle r\geq 2}$. (Hopf defined this for the special case ${\displaystyle q=r+1}$.)

The J-homomorphism can be defined as follows. An element of the special orthogonal group SO(q) can be regarded as a map

${\displaystyle S^{q-1}\rightarrow S^{q-1}}$

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

${\displaystyle S^{r}\times S^{q-1}\rightarrow S^{q-1}}$

Applying the Hopf construction to this gives a map

${\displaystyle S^{r+q}=S^{r}*S^{q-1}\rightarrow S(S^{q-1})=S^{q}}$

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

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

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

where ${\displaystyle \mathrm {SO} }$ 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 ${\displaystyle \pi _{r}(\operatorname {SO} )}$ 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 ${\displaystyle \pi _{r}^{S}}$ 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 ${\displaystyle \mathbb {Q} /\mathbb {Z} }$. 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 ${\displaystyle B_{2n}/4n}$, where ${\displaystyle B_{2n}}$ is a Bernoulli number. In the remaining cases where r is 2, 4, 5, or 6 mod 8 the image is trivial because ${\displaystyle \pi _{r}(\operatorname {SO} )}$ is trivial.

r	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17
${\displaystyle \pi _{r}(\operatorname {SO} )}$	1	2	1	${\displaystyle \mathbb {Z} }$	1	1	1	${\displaystyle \mathbb {Z} }$	2	2	1	${\displaystyle \mathbb {Z} }$	1	1	1	${\displaystyle \mathbb {Z} }$	2	2
${\displaystyle |\operatorname {im} (J)|}$	1	2	1	24	1	1	1	240	2	2	1	504	1	1	1	480	2	2
${\displaystyle \pi _{r}^{S}}$	${\displaystyle \mathbb {Z} }$	2	2	24	1	1	2	240	2^2	2^3	6	504	1	3	2^2	480×2	2^2	2^4
${\displaystyle B_{2n}}$				^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 ${\displaystyle J\colon \pi _{n}(\mathrm {SO} )\to \pi _{n}^{S}}$ appears in the group Θ_n of h-cobordism classes of oriented homotopy n-spheres (Kosinski (1992)).

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, 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", Topology, 3 (2): 137, doi:10.1016/0040-9383(65)90040-6
• Adams, J. F. (1965b), "On the groups J(X) III", Topology, 3 (3): 193, doi:10.1016/0040-9383(65)90054-6
• Adams, J. F. (1966), "On the groups J(X) IV", Topology, 5: 21, doi:10.1016/0040-9383(66)90004-8. "Correction", Topology, 7 (3): 331, 1968, doi:10.1016/0040-9383(68)90010-4
• Hopf, Heinz (1935), "Über die Abbildungen von Sphären auf Sphäre niedrigerer Dimension", Fundamenta Mathematicae, 25: 427–440
• Kosinski, Antoni A. (1992), Differential Manifolds, San Diego, CA: Academic Press, pp. 195ff, ISBN 0-12-421850-4
• Milnor, John W. (2011), "Differential topology forty-six years later" (PDF), Notices of the American Mathematical Society, 58 (6): 804–809
• Quillen, Daniel (1971), "The Adams conjecture", Topology, 10: 67–80, doi:10.1016/0040-9383(71)90018-8, MR 0279804
• Switzer, Robert M. (1975), 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, JSTOR 1968956, MR 0007107
• Whitehead, George W. (1978), Elements of homotopy theory, Berlin: Springer, ISBN 0-387-90336-4, MR 0516508