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====References====
 
====References====
<table><TR><TD valign="top">[1a]</TD> <TD valign="top"> G.W. Whitehead,   "On the homotopy groups of spheres and rotation groups" ''Ann. of Math.'' , '''43''' (1942) pp. 634–640</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top"> G.W. Whitehead,   "A generalization of the Hopf invariant" ''Ann. of Math.'' , '''51''' (1950) pp. 192–237</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> R. Bott,   "The stable homotopy of the classical groups" ''Ann. of Math.'' , '''70''' (1959) pp. 313–337</TD></TR><TR><TD valign="top">[3a]</TD> <TD valign="top"> J.F. Adams,   "On the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778049.png" /> - I" ''Topology'' , '''2''' (1963) pp. 181–195</TD></TR><TR><TD valign="top">[3b]</TD> <TD valign="top"> J.F. Adams,   "On the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778050.png" /> - II" ''Topology'' , '''3''' (1965) pp. 137–171</TD></TR><TR><TD valign="top">[3c]</TD> <TD valign="top"> J.F. Adams,   "On the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778051.png" /> - III" ''Topology'' , '''3''' (1965) pp. 193–222</TD></TR><TR><TD valign="top">[3d]</TD> <TD valign="top"> J.F. Adams,   "On the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778052.png" /> - IV" ''Topology'' , '''5''' (1966) pp. 21–71</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> J.C. Becker,   D.H. Gottlieb,   "The transfer map and fiber bundles" ''Topology'' , '''14''' (1975) pp. 1–12</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> J.F. Adams,   "Infinite loop spaces" , Princeton Univ. Press (1978)</TD></TR></table>
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<table><TR><TD valign="top">[1a]</TD> <TD valign="top"> G.W. Whitehead, "On the homotopy groups of spheres and rotation groups" ''Ann. of Math.'' , '''43''' (1942) pp. 634–640 {{MR|0007107}} {{ZBL|0060.41105}} </TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top"> G.W. Whitehead, "A generalization of the Hopf invariant" ''Ann. of Math.'' , '''51''' (1950) pp. 192–237 {{MR|0041435}} {{ZBL|0045.44202}} {{ZBL|0041.51903}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> R. Bott, "The stable homotopy of the classical groups" ''Ann. of Math.'' , '''70''' (1959) pp. 313–337 {{MR|0110104}} {{ZBL|0129.15601}} </TD></TR><TR><TD valign="top">[3a]</TD> <TD valign="top"> J.F. Adams, "On the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778049.png" /> - I" ''Topology'' , '''2''' (1963) pp. 181–195</TD></TR><TR><TD valign="top">[3b]</TD> <TD valign="top"> J.F. Adams, "On the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778050.png" /> - II" ''Topology'' , '''3''' (1965) pp. 137–171</TD></TR><TR><TD valign="top">[3c]</TD> <TD valign="top"> J.F. Adams, "On the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778051.png" /> - III" ''Topology'' , '''3''' (1965) pp. 193–222</TD></TR><TR><TD valign="top">[3d]</TD> <TD valign="top"> J.F. Adams, "On the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778052.png" /> - IV" ''Topology'' , '''5''' (1966) pp. 21–71</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> J.C. Becker, D.H. Gottlieb, "The transfer map and fiber bundles" ''Topology'' , '''14''' (1975) pp. 1–12 {{MR|0377873}} {{ZBL|0306.55017}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> J.F. Adams, "Infinite loop spaces" , Princeton Univ. Press (1978) {{MR|0505692}} {{ZBL|0398.55008}} </TD></TR></table>
  
  
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> B. Gray,   "Homotopy theory. An introduction to algebraic topology" , Acad. Press (1975) pp. 334</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> R.M. Switzer,   "Algebraic topology - homotopy and homology" , Springer (1975) pp. 480ff</TD></TR></table>
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> B. Gray, "Homotopy theory. An introduction to algebraic topology" , Acad. Press (1975) pp. 334 {{MR|0402714}} {{ZBL|0322.55001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> R.M. Switzer, "Algebraic topology - homotopy and homology" , Springer (1975) pp. 480ff {{MR|0385836}} {{ZBL|0305.55001}} </TD></TR></table>

Revision as of 10:54, 1 April 2012

-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 :'

<tbody> </tbody>
14 14
7 4

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
How to Cite This Entry:
Whitehead homomorphism. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Whitehead_homomorphism&oldid=24173
This article was adapted from an original article by A.V. Shokurov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article