J-homomorphism

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Short description: From a homotopy group of a special orthogonal group to a homotopy group of spheres

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

[math]\displaystyle{ J \colon \pi_r (\mathrm{SO}(q)) \to \pi_{r+q}(S^q) }[/math]

of abelian groups for integers q, and [math]\displaystyle{ r \ge 2 }[/math]. (Hopf defined this for the special case [math]\displaystyle{ q = r+1 }[/math].)

The J-homomorphism can be defined as follows. An element of the special orthogonal group SO(q) can be regarded as a map

[math]\displaystyle{ S^{q-1}\rightarrow S^{q-1} }[/math]

and the homotopy group [math]\displaystyle{ \pi_r(\operatorname{SO}(q)) }[/math]) consists of homotopy classes of maps from the r-sphere to SO(q). Thus an element of [math]\displaystyle{ \pi_r(\operatorname{SO}(q)) }[/math] can be represented by a map

[math]\displaystyle{ S^r\times S^{q-1}\rightarrow S^{q-1} }[/math]

Applying the Hopf construction to this gives a map

[math]\displaystyle{ S^{r+q}= S^r*S^{q-1}\rightarrow S( S^{q-1}) =S^q }[/math]

in [math]\displaystyle{ \pi_{r+q}(S^q) }[/math], which Whitehead defined as the image of the element of [math]\displaystyle{ \pi_r(\operatorname{SO}(q)) }[/math] under the J-homomorphism.

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

[math]\displaystyle{ J \colon \pi_r(\mathrm{SO}) \to \pi_r^S , }[/math]

where [math]\displaystyle{ \mathrm{SO} }[/math] 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 [math]\displaystyle{ \pi_r(\operatorname{SO}) }[/math] 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 modulo 4, and order 1 otherwise (Switzer 1975). In particular the image of the stable J-homomorphism is cyclic. The stable homotopy groups [math]\displaystyle{ \pi_r^S }[/math] 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 [math]\displaystyle{ \Q/\Z }[/math]. 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 mod 4, the image is a cyclic group of order equal to the denominator of [math]\displaystyle{ B_{2n}/4n }[/math], where [math]\displaystyle{ B_{2n} }[/math] is a Bernoulli number. In the remaining cases where r is 2, 4, 5, or 6 mod 8 the image is trivial because [math]\displaystyle{ \pi_r(\operatorname{SO}) }[/math] is trivial.

r 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
[math]\displaystyle{ \pi_r(\operatorname{SO}) }[/math] 1 2 1 [math]\displaystyle{ \Z }[/math] 1 1 1 [math]\displaystyle{ \Z }[/math] 2 2 1 [math]\displaystyle{ \Z }[/math] 1 1 1 [math]\displaystyle{ \Z }[/math] 2 2
[math]\displaystyle{ |\operatorname{im}(J)| }[/math] 1 2 1 24 1 1 1 240 2 2 1 504 1 1 1 480 2 2
[math]\displaystyle{ \pi_r^S }[/math] [math]\displaystyle{ \Z }[/math] 2 2 24 1 1 2 240 22 23 6 504 1 3 22 480×2 22 24
[math]\displaystyle{ B_{2n} }[/math] 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 [math]\displaystyle{ J \colon \pi_n(\mathrm{SO}) \to \pi_n^S }[/math] appears in the group Θn of h-cobordism classes of oriented homotopy n-spheres ((Kosinski 1992)).

References