Whitehead homomorphism

-homomorphism

A homomorphism from the stable homotopy group of the spectrum of to the stable homotopy group of the spectrum of the sphere , defined in a special way. One construction of the Whitehead group is by the Hopf construction: A mapping determines a mapping , which can be extended to a mapping of to the upper hemi-sphere of . There is also an extension to the lower hemi-sphere of , and this determines a mapping . This construction gives a mapping of homotopy classes, and so defines a homomorphism , 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, , for the following values of and :

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The stable homotopy groups are described by the Bott periodicity theorem [2]:

<tbody> </tbody> 0 1 2 3 4 5 6 7 0 0 0 0

The image of the Whitehead homomorphism has been completely calculated (cf. [4], [5]): for () and the Whitehead group is a monomorphism and its image is a direct summand in the group ; for () and the Whitehead group is a monomorphism on a direct summand of ; for the image of the Whitehead group is the cyclic group of order , giving a direct summand in , where is the denominator of the irreducible fraction , being the -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 of topological spaces, quite generally the Hopf construction gives a mapping

from the join of and to the suspension of , as follows. Consider

The join is a certain quotient space of and is a quotient space of . An elementary check shows that is compatible with the corresponding equivalence relations, and hence defines a mapping as desired. Recall that , cf. Join.

Now, let be a mapping; each element of induces a mapping of the -sphere into itself. Hence induces a mapping

Applying the Hopf construction to gives the mapping :

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