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]:
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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 ![]() |
[3b] | J.F. Adams, "On the groups ![]() |
[3c] | J.F. Adams, "On the groups ![]() |
[3d] | J.F. Adams, "On the groups ![]() |
[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 |
Whitehead homomorphism. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Whitehead_homomorphism&oldid=49208