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

J \colon \pi_r (\mathrm{SO}(q)) \to \pi_{r+q}(S^q) \,\!

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

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

and the homotopy group π_r(SO(q)) consists of homotopy-equivalence classes of maps from the r-sphere to SO(q). Thus an element of π_r(SO(q)) can be represented by a map

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

Applying the Hopf construction to this gives a map

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

in π_r+q(S^q), which Whitehead defined as the image of the element of π_r(SO(q)) under the J-homomorphism.

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

J \colon \pi_r(\mathrm{SO}) \to \pi_r^S , \,\!

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 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 (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 = 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
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	2²	2³	6	504	1	3	2²	480×2	2²	2⁴
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", 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", 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