##### Actions

$J$- 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 [2]:

<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. [4], [5]): 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).

#### References

 [1a] G.W. Whitehead, "On the homotopy groups of spheres and rotation groups" Ann. of Math. , 43 (1942) pp. 634–640 MR0007107 Zbl 0060.41105 [1b] G.W. Whitehead, "A generalization of the Hopf invariant" Ann. of Math. , 51 (1950) pp. 192–237 MR0041435 Zbl 0045.44202 Zbl 0041.51903 [2] R. Bott, "The stable homotopy of the classical groups" Ann. of Math. , 70 (1959) pp. 313–337 MR0110104 Zbl 0129.15601 [3a] J.F. Adams, "On the groups - I" Topology , 2 (1963) pp. 181–195 [3b] J.F. Adams, "On the groups - II" Topology , 3 (1965) pp. 137–171 [3c] J.F. Adams, "On the groups - III" Topology , 3 (1965) pp. 193–222 [3d] J.F. Adams, "On the groups - IV" Topology , 5 (1966) pp. 21–71 [4] J.C. Becker, D.H. Gottlieb, "The transfer map and fiber bundles" Topology , 14 (1975) pp. 1–12 MR0377873 Zbl 0306.55017 [5] J.F. Adams, "Infinite loop spaces" , Princeton Univ. Press (1978) MR0505692 Zbl 0398.55008

Given a mapping $f: X \times Y \rightarrow Z$ of topological spaces, quite generally the Hopf construction gives a mapping

$$\Gamma f : X \star Y \rightarrow SZ$$

from the join $X \star Y$ of $X$ and $Y$ to the suspension $SZ$ of $Z$, as follows. Consider

$$f \times \mathop{\rm id} : X \times Y \times I \rightarrow Z \times I,\ \ ( x, y, t) \mapsto ( f( x, y), t).$$

The join $X \star Y$ is a certain quotient space of $X \times Y \times Z$ and $SZ$ is a quotient space of $Z \times I$. An elementary check shows that $f \times \mathop{\rm id}$ is compatible with the corresponding equivalence relations, and hence defines a mapping $\Gamma f$ as desired. Recall that $S ^ {m} \star S ^ {n} \cong S ^ {m+} n+ 1$, cf. Join.

Now, let $\phi : S ^ {m} \rightarrow \mathop{\rm SO} ( q)$ be a mapping; each element of $\mathop{\rm SO} ( q)$ induces a mapping $S ^ {q-} 1 \rightarrow S ^ {q-} 1$ of the $( q- 1)$- sphere into itself. Hence $\phi$ induces a mapping

$$\widetilde \phi : S ^ {m} \times S ^ {q-} 1 \rightarrow S ^ {q-} 1 .$$

Applying the Hopf construction to $\widetilde \phi$ gives the mapping $J \phi$:

$$S ^ {m+} q \cong S ^ {m} \star S ^ {q-} 1 \mathop \rightarrow \limits ^ { {\Gamma \widetilde \phi }} \ S( S ^ {q-} 1 ) \cong S ^ {q} .$$

#### References

 [a1] B. Gray, "Homotopy theory. An introduction to algebraic topology" , Acad. Press (1975) pp. 334 MR0402714 Zbl 0322.55001 [a2] R.M. Switzer, "Algebraic topology - homotopy and homology" , Springer (1975) pp. 480ff MR0385836 Zbl 0305.55001
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