J-homomorphism

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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[edit]

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 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

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 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 \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[edit]

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 16 130 142 130

Applications[edit]

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[edit]