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$J$- homomorphism

A homomorphism from the stable homotopy group of the spectrum of $\mathop{\rm SO}$ to the stable homotopy group of the spectrum of the sphere $S ^ {0}$, defined in a special way. One construction of the Whitehead group is by the Hopf construction: A mapping $\phi : S ^ {m} \rightarrow \mathop{\rm SO} ( q)$ determines a mapping $( J \phi ) : S ^ {m} \times S ^ {q-} 1 \rightarrow S ^ {q-} 1$, which can be extended to a mapping $J \phi : S ^ {m} \times E ^ {q} \rightarrow E _ {+} ^ {q}$ of $S ^ {m} \times E ^ {q}$ to the upper hemi-sphere of $S ^ {q}$. There is also an extension $J \phi : E ^ {m+} 1 \times S ^ {q-} 1 \rightarrow E _ {-} ^ {q}$ to the lower hemi-sphere of $S ^ {q}$, and this determines a mapping $J \phi : S ^ {m+} q \rightarrow S ^ {q}$. This construction gives a mapping of homotopy classes, and so defines a homomorphism $J: \pi _ {m} ^ {S} ( \mathop{\rm SO} ) \rightarrow \pi _ {m} ^ {S} ( S ^ {0} )$, called the Whitehead group.

This homomorphism was first constructed by G.W. Whitehead , who also proved a theorem on the non-triviality of the infinite series of homotopy groups of spheres, $\pi _ {n} ( S ^ {r} ) \neq 0$, for the following values of $n$ and $r$:

<tbody> </tbody>
 $n$ 14 14 $8k$ $16k+ 2$ $8k+ 1$ $16k+ 3$ $r$ 7 4 $4k$ $8k$ $4k+ 1$ $8k+ 1$

The stable homotopy groups $\pi _ {m} ^ {S} ( \mathop{\rm SO} )$ are described by the Bott periodicity theorem :

<tbody> </tbody>
 $m \mathop{\rm mod} 8$ 0 1 2 3 4 5 6 7 $\pi _ {m} ^ {S} ( \mathop{\rm SO} )$ $\mathbf Z _ {2}$ $\mathbf Z _ {2}$ 0 $\mathbf Z$ 0 0 0 $\mathbf Z$

The image of the Whitehead homomorphism has been completely calculated (cf. , ): for $m \equiv 0$( $\mathop{\rm mod} 8$) and $m> 0$ the Whitehead group is a monomorphism and its image is a direct summand in the group $\pi _ {m} ^ {S} ( S ^ {0} )$; for $m\equiv 1$( $\mathop{\rm mod} 8$) and $m> 1$ the Whitehead group is a monomorphism on a direct summand of $\pi _ {m} ^ {S} ( S ^ {0} )$; for $m= 4s- 1$ the image of the Whitehead group is the cyclic group of order $\tau ( 2s)$, giving a direct summand in $\pi _ {m} ^ {S} ( S ^ {0} )$, where $\tau ( 2s)$ is the denominator of the irreducible fraction $B _ {s} /( 4s)$, $B _ {s}$ being the $s$- th Bernoulli number (cf. Bernoulli numbers).

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