J-homomorphism

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

:$J\; \backslash colon\; \backslash pi\_r\; (\backslash mathrm\{SO\}(q))\; \backslash to\; \backslash pi\_\{r+q\}(S^q)$

of abelian groups for integers q, and $r\; \backslash ge\; 2$. (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

:$S^\{q-1\}\backslash rightarrow\; S^\{q-1\}$

and the homotopy group $\backslash pi\_r(\backslash operatorname\{SO\}(q))$) consists of homotopy classes of maps from the r-sphere to SO(q).

Thus an element of $\backslash pi\_r(\backslash operatorname\{SO\}(q))$ can be represented by a map

:$S^r\backslash times\; S^\{q-1\}\backslash rightarrow\; S^\{q-1\}$

Applying the Hopf construction to this gives a map

:$S^\{r+q\}=\; S^r*S^\{q-1\}\backslash rightarrow\; S(\; S^\{q-1\})\; =S^q$

in $\backslash pi\_\{r+q\}(S^q)$, which Whitehead defined as the image of the element of $\backslash pi\_r(\backslash operatorname\{SO\}(q))$ under the J-homomorphism.

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

:$J\; \backslash colon\; \backslash pi\_r(\backslash mathrm\{SO\})\; \backslash to\; \backslash pi\_r^S\; ,$

where $\backslash mathrm\{SO\}$ is the infinite special orthogonal group, and the right-hand side is the r-th stable stem of the stable homotopy groups of spheres.

The image of the J-homomorphism was described by {{harvs|txt|last=Adams|first=Frank|author-link=Frank Adams|year=1966}}, assuming the Adams conjecture of {{harvtxt|Adams|1963}} which was proved by {{harvs|txt|last=Quillen|first=Daniel|author-link=Daniel Quillen|year=1971}}, as follows. The group $\backslash pi\_r(\backslash operatorname\{SO\})$ 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 {{harv|Switzer|1975|p=488}}. In particular the image of the stable J-homomorphism is cyclic. The stable homotopy groups $\backslash pi\_r^S$ are the direct sum of the (cyclic) image of the J-homomorphism, and the kernel of the Adams e-invariant {{harv|Adams|1966}}, a homomorphism from the stable homotopy groups to $\backslash Q/\backslash Z$. 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 or 7 mod 8, the image is a cyclic group of order equal to the denominator of $B\_\{2n\}/4n$, where $B\_\{2n\}$ is a Bernoulli number. In the remaining cases where r is 2, 4, 5, or 6 mod 8 the image is trivial because $\backslash pi\_r(\backslash operatorname\{SO\})$ is trivial.

:

{{harvs|txt|last=Atiyah | first=Michael | author-link=Michael Atiyah |year=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 $J\; \backslash colon\; \backslash pi\_n(\backslash mathrm\{SO\})\; \backslash to\; \backslash pi\_n^S$ appears in the group Θ_n of h-cobordism classes of oriented homotopy n-spheres ({{harvtxt|Kosinski |1992}}).

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• J-homomorphism at the nLab

Category:Homotopy theory

Category:Topology of Lie groups