J Homomorphism

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

:$J \colon \pi_r (\mathrm(q)) \to \pi_(S^q)$

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
:$S^\rightarrow S^$
and the homotopy group $\pi_r(\operatorname(q))$) consists of homotopy classes of maps from the ''r''-sphere to SO(''q'').
Thus an element of $\pi_r(\operatorname(q))$ can be represented by a map
:$S^r\times S^\rightarrow S^$
Applying the Hopf construction to this gives a map
:$S^= S^r*S^\rightarrow S( S^) =S^q$
in $\pi_(S^q)$, which Whitehead defined as the image of the element of $\pi_r(\operatorname(q))$ under the J-homomorphism.

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

:$J \colon \pi_r(\mathrm) \to \pi_r^S ,$

where $\mathrm$ 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 , assuming the Adams conjecture of which was proved by , as follows. The group $\pi_r(\operatorname)$ 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 . In particular the image of the stable ''J''-homomorphism is cyclic. The stable homotopy groups $\pi_r^S$ 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 mod 4, the image is a cyclic group of order equal to the denominator of $B_/4n$, where $B_$ is a Bernoulli number. In the remaining cases where ''r'' is 2, 4, 5, or 6 mod 8 the image is trivial because $\pi_r(\operatorname)$ is trivial.

:

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 $J \colon \pi_n(\mathrm) \to \pi_n^S$ appears in the group Θ_''n'' of ''h''-cobordism classes of oriented homotopy ''n''-spheres ().

References

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* {{Citation , last=Whitehead , first=George W. , author-link=George W. Whitehead, title=Elements of homotopy theory , publisher=Springer , location=Berlin , year=1978 , isbn=0-387-90336-4 , mr= 0516508

Homotopy theory
Topology of Lie groups