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 Heinz Hopf (1935).

Definition

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

 

of abelian groups for integers q, and  . (Hopf defined this for the special case  .)

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

 

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

 

Applying the Hopf construction to this gives a map

 

in  , which Whitehead defined as the image of the element of   under the J-homomorphism.

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

 

where   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

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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   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   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  , where   is a Bernoulli number. In the remaining cases where r is 2, 4, 5, or 6 mod 8 the image is trivial because   is trivial.

r 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
  1 2 1   1 1 1   2 2 1   1 1 1   2 2
  1 2 1 24 1 1 1 240 2 2 1 504 1 1 1 480 2 2
    2 2 24 1 1 2 240 22 23 6 504 1 3 22 480×2 22 24
  16 130 142 130

Applications

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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   appears in the group Θn of h-cobordism classes of oriented homotopy n-spheres (Kosinski (1992)).

References

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