J-homomorphism

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In mathematics, the J-homomorphism is a mapping from the homotopy groups of the special orthogonal groups to the homotopy groups of spheres, defined by George W. Whitehead.

The original homomorphism is defined geometrically, and gives a homomorphism

of abelian groups (for integers q, and r ≥ 2).

The stable J-homomorphism in stable homotopy theory gives a homomorphism

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.

The image of the J-homomorphism was described by Adams (1966) and Quillen (1971) as follows. The group π_r(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 mod 8, infinite if r is 3 mod 4, and order 1 otherwise. 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 = 4n−1 is 3 mod 4 and positive 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 π_r(SO) is trivial.

r	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17
πr(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
πrS	Z	2	2	24	1	1	2	240	22	23	6	504	1	3	22	480×2	22	24
B2n				1⁄6				−1⁄30				1⁄42				−1⁄30

An important step in the theory was the Adams conjecture from 1963, which allowed the order of the image of the stable J-homomorphism to be determined (it is cyclic; see Switzer (1975), p. 488) for details). Frank Adams's conjecture was proved about eight years later by Daniel Quillen. The cokernel of the J-homomorphism is of interest for counting exotic spheres.

[edit] References

• Adams, J. F. (1963), “On the groups J(X) I”, Topology 2, DOI 10.1016/0040-9383(63)90001-6 
• Adams, J. F. (1965a), “On the groups J(X) II”, Topology 3, DOI 10.1016/0040-9383(65)90040-6 
• Adams, J. F. (1965b), “On the groups J(X) III”, Topology 3, DOI 10.1016/0040-9383(65)90054-6 
• Adams, J. F. (1966), “On the groups J(X) IV”, Topology 5, DOI 10.1016/0040-9383(66)90004-8  “Correction”, Topology 7, 1968, DOI 10.1016/0040-9383(68)90010-4 
• Quillen, D. (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. (1978). Elements of homotopy theory. Berlin: Springer. MR0516508. ISBN 0-387-90336-4.