J-homomorphism explained

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, extending a construction of .

Definition

Whitehead's original homomorphism is defined geometrically, and gives a homomorphism

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

of abelian groups for integers q, and

r \ge 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} → S ^{q-1} and the homotopy group \pi _{r(\operatorname{SO}(q))} ) consists of homotopy classes of maps from the r-sphere to SO(q).Thus an element of \pi _{r(\operatorname{SO}(q))} can be represented by a mapS ^{r x} S ^{q-1} → S ^{q-1} Applying the Hopf construction to this gives a mapS ^{r+q} = S ^{r*S} ^{q-1} → S( S ^{q-1} ) = S ^{q} in \pi _{r+q} (S ^{q)} , which Whitehead defined as the image of the element of \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:

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

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

\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 . In particular the image of the stable J-homomorphism is cyclic. The stable homotopy groups \pi ^{S} _{r} are the direct sum of the (cyclic) image of the J-homomorphism, and the kernel of the Adams e-invariant, a homomorphism from the stable homotopy groups to \Q/\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 B _{2n} / 4 n , 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 \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
\pi _{r(\operatorname{SO})}	1	2	1	\Z	1	1	1	\Z	2	2	1	\Z	1	1	1	\Z	2	2
	\operatorname(J)		1	2	1	24	1	1	1	240	2	2	1	504	1	1	1	480	2	2
\pi ^{S} _{r}	\Z	2	2	24	1	1	2	240	2^2	2^3	6	504	1	3	2^2	480×2	2^2	2^4
B _{2n}				^1⁄6				−^1⁄30				^1⁄42				−^1⁄30

Applications

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

J \colon \pi _{n(SO)} \to \pi ^{S} _{n} appears in the group Θ_n of h-cobordism classes of oriented homotopy n-spheres .

References

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