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 Hopf (1935).
Definition
Whitehead's original homomorphism is defined geometrically, and gives a homomorphism
of abelian groups for integers q, and r ≥ 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
and the homotopy group πr(SO(q)) consists of homotopy-equivalence classes of maps from the r-sphere to SO(q). Thus an element of πr(SO(q)) can be represented by a map
Applying the Hopf construction to this gives a map
in πr+q(Sq), which Whitehead defined as the image of the element of πr(SO(q)) 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 Adams (1966), assuming the Adams conjecture of Adams (1963) which was proved by Quillen (1971), as follows. The group πr(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 (Switzer 1975, p. 488). In particular the image of the stable J-homomorphism is cyclic. The stable homotopy groups πrS 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 B2n/4n, where B2n is a Bernoulli number. In the remaining cases where r is 2, 4, 5, or 6 mod 8 the image is trivial because πr(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
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", 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 Adams, J (1968), "Correction", Topology, 7 (3): 331, 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, ISSN 0016-2736
- 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. an International Journal of Mathematics, 10: 67–80, doi:10.1016/0040-9383(71)90018-8, ISSN 0040-9383, 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, ISSN 0003-486X, JSTOR 1968956, MR 0007107
- Whitehead, George W. (1978), Elements of homotopy theory, Berlin: Springer, ISBN 0-387-90336-4, MR 0516508