Whitehead homomorphism

$ 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

Comments

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