View source for Whitehead homomorphism

''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w0977802.png" />-homomorphism''

A homomorphism from the [[Stable homotopy group|stable homotopy group]] of the spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w0977803.png" /> to the stable homotopy group of the spectrum of the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w0977804.png" />, defined in a special way. One construction of the Whitehead group is by the Hopf construction: A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w0977805.png" /> determines a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w0977806.png" />, which can be extended to a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w0977807.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w0977808.png" /> to the upper hemi-sphere of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w0977809.png" />. There is also an extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778010.png" /> to the lower hemi-sphere of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778011.png" />, and this determines a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778012.png" />. This construction gives a mapping of homotopy classes, and so defines a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778013.png" />, 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, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778014.png" />, for the following values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778016.png" />:''''''<table border="0" cellpadding="0" cellspacing="0" style="background-color:black;"> <tr><td> <table border="0" cellspacing="1" cellpadding="4" style="background-color:black;"> <tbody> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778017.png" /></td> <td colname="2" style="background-color:white;" colspan="1">14</td> <td colname="3" style="background-color:white;" colspan="1">14</td> <td colname="4" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778018.png" /></td> <td colname="5" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778019.png" /></td> <td colname="6" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778020.png" /></td> <td colname="7" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778021.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778022.png" /></td> <td colname="2" style="background-color:white;" colspan="1">7</td> <td colname="3" style="background-color:white;" colspan="1">4</td> <td colname="4" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778023.png" /></td> <td colname="5" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778024.png" /></td> <td colname="6" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778025.png" /></td> <td colname="7" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778026.png" /></td> </tr> </tbody> </table>

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The stable homotopy groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778027.png" /> are described by the Bott periodicity theorem [[#References|[2]]]:''''''<table border="0" cellpadding="0" cellspacing="0" style="background-color:black;"> <tr><td> <table border="0" cellspacing="1" cellpadding="4" style="background-color:black;"> <tbody> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778028.png" /></td> <td colname="2" style="background-color:white;" colspan="1">0</td> <td colname="3" style="background-color:white;" colspan="1">1</td> <td colname="4" style="background-color:white;" colspan="1">2</td> <td colname="5" style="background-color:white;" colspan="1">3</td> <td colname="6" style="background-color:white;" colspan="1">4</td> <td colname="7" style="background-color:white;" colspan="1">5</td> <td colname="8" style="background-color:white;" colspan="1">6</td> <td colname="9" style="background-color:white;" colspan="1">7</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778029.png" /></td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778030.png" /></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778031.png" /></td> <td colname="4" style="background-color:white;" colspan="1">0</td> <td colname="5" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778032.png" /></td> <td colname="6" style="background-color:white;" colspan="1">0</td> <td colname="7" style="background-color:white;" colspan="1">0</td> <td colname="8" style="background-color:white;" colspan="1">0</td> <td colname="9" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778033.png" /></td> </tr> </tbody> </table>

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The image of the Whitehead homomorphism has been completely calculated (cf. [[#References|[4]]], [[#References|[5]]]): for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778034.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778035.png" />) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778036.png" /> the Whitehead group is a monomorphism and its image is a direct summand in the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778037.png" />; for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778038.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778039.png" />) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778040.png" /> the Whitehead group is a monomorphism on a direct summand of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778041.png" />; for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778042.png" /> the image of the Whitehead group is the cyclic group of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778043.png" />, giving a direct summand in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778044.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778045.png" /> is the denominator of the irreducible fraction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778047.png" /> being the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778048.png" />-th Bernoulli number (cf. [[Bernoulli numbers|Bernoulli numbers]]).

====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>



====Comments====
Given a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778053.png" /> of topological spaces, quite generally the Hopf construction gives a mapping

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778054.png" /></td> </tr></table>

from the [[Join|join]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778055.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778056.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778057.png" /> to the [[Suspension|suspension]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778058.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778059.png" />, as follows. Consider

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778060.png" /></td> </tr></table>

The join <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778061.png" /> is a certain quotient space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778062.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778063.png" /> is a quotient space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778064.png" />. An elementary check shows that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778065.png" /> is compatible with the corresponding equivalence relations, and hence defines a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778066.png" /> as desired. Recall that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778067.png" />, cf. [[Join|Join]].

Now, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778068.png" /> be a mapping; each element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778069.png" /> induces a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778070.png" /> of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778071.png" />-sphere into itself. Hence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778072.png" /> induces a mapping

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778073.png" /></td> </tr></table>

Applying the Hopf construction to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778074.png" /> gives the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778075.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w09778076.png" /></td> </tr></table>

====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>