J-homomorphism

From The Right Wiki
(Redirected from E-invariant)
Jump to navigationJump to search

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

J:πr(SO(q))πr+q(Sq)

of abelian groups for integers q, and r2. (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

Sq1Sq1

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

Sr×Sq1Sq1

Applying the Hopf construction to this gives a map

Sr+q=Sr*Sq1S(Sq1)=Sq

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:

J:πr(SO)πrS,

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 Frank Adams (1966), assuming the Adams conjecture of Adams (1963) which was proved by Daniel 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 modulo 8, infinite if r is 3 or 7 modulo 8, 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 /. 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 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 1 1 1 2 2 1 1 1 1 2 2
|im(J)| 1 2 1 24 1 1 1 240 2 2 1 504 1 1 1 480 2 2
πrS 2 2 24 1 1 2 240 22 23 6 504 1 3 22 480×2 22 24
B2n 16 130 142 130

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 J:πn(SO)πnS appears in the group Θn of h-cobordism classes of oriented homotopy n-spheres (Kosinski (1992)).

References