J-homomorphism

Short description: From a homotopy group of a special orthogonal group to a homotopy group of spheres

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

[math]\displaystyle{ J \colon \pi_r (\mathrm{SO}(q)) \to \pi_{r+q}(S^q) }[/math]

of abelian groups for integers q, and [math]\displaystyle{ r \ge 2 }[/math]. (Hopf defined this for the special case [math]\displaystyle{ q = r+1 }[/math].)

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

[math]\displaystyle{ S^{q-1}\rightarrow S^{q-1} }[/math]

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

[math]\displaystyle{ S^r\times S^{q-1}\rightarrow S^{q-1} }[/math]

Applying the Hopf construction to this gives a map

[math]\displaystyle{ S^{r+q}= S^r*S^{q-1}\rightarrow S( S^{q-1}) =S^q }[/math]

in [math]\displaystyle{ \pi_{r+q}(S^q) }[/math], which Whitehead defined as the image of the element of [math]\displaystyle{ \pi_r(\operatorname{SO}(q)) }[/math] under the J-homomorphism.

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

[math]\displaystyle{ J \colon \pi_r(\mathrm{SO}) \to \pi_r^S , }[/math]

where [math]\displaystyle{ \mathrm{SO} }[/math] 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 [math]\displaystyle{ \pi_r(\operatorname{SO}) }[/math] 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). In particular the image of the stable J-homomorphism is cyclic. The stable homotopy groups [math]\displaystyle{ \pi_r^S }[/math] 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 [math]\displaystyle{ \Q/\Z }[/math]. 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 [math]\displaystyle{ B_{2n}/4n }[/math], where [math]\displaystyle{ B_{2n} }[/math] is a Bernoulli number. In the remaining cases where r is 2, 4, 5, or 6 mod 8 the image is trivial because [math]\displaystyle{ \pi_r(\operatorname{SO}) }[/math] is trivial.

r	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17
[math]\displaystyle{ \pi_r(\operatorname{SO}) }[/math]	1	2	1	[math]\displaystyle{ \Z }[/math]	1	1	1	[math]\displaystyle{ \Z }[/math]	2	2	1	[math]\displaystyle{ \Z }[/math]	1	1	1	[math]\displaystyle{ \Z }[/math]	2	2
[math]\displaystyle{ |\operatorname{im}(J)| }[/math]	1	2	1	24	1	1	1	240	2	2	1	504	1	1	1	480	2	2
[math]\displaystyle{ \pi_r^S }[/math]	[math]\displaystyle{ \Z }[/math]	2	2	24	1	1	2	240	22	23	6	504	1	3	22	480×2	22	24
[math]\displaystyle{ B_{2n} }[/math]				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 [math]\displaystyle{ J \colon \pi_n(\mathrm{SO}) \to \pi_n^S }[/math] 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 
• 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, http://matwbn.icm.edu.pl/tresc.php?wyd=1&tom=25 
• Kosinski, Antoni A. (1992), Differential Manifolds, San Diego, CA: Academic Press, pp. 195ff, ISBN 0-12-421850-4, https://archive.org/details/differentialmani0000kosi/page/195 
• Milnor, John W. (2011), "Differential topology forty-six years later", Notices of the American Mathematical Society 58 (6): 804–809, https://www.ams.org/notices/201106/rtx110600804p.pdf 
• Quillen, Daniel (1971), "The Adams conjecture", Topology 10: 67–80, doi:10.1016/0040-9383(71)90018-8 
• 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 
• Whitehead, George W. (1978), Elements of homotopy theory, Berlin: Springer, ISBN 0-387-90336-4