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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097780/w0977802.png" />-homomorphism''
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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.
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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" />:
+
'' $  J $-
 +
homomorphism''
  
<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>
+
A homomorphism from the [[Stable homotopy group|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 $:
 +
 
 +
<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"> $  n $
 +
</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"> $  8k $
 +
</td> <td colname="5" style="background-color:white;" colspan="1"> $  16k+ 2 $
 +
</td> <td colname="6" style="background-color:white;" colspan="1"> $  8k+ 1 $
 +
</td> <td colname="7" style="background-color:white;" colspan="1"> $  16k+ 3 $
 +
</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  r $
 +
</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"> $  4k $
 +
</td> <td colname="5" style="background-color:white;" colspan="1"> $  8k $
 +
</td> <td colname="6" style="background-color:white;" colspan="1"> $  4k+ 1 $
 +
</td> <td colname="7" style="background-color:white;" colspan="1"> $  8k+ 1 $
 +
</td> </tr> </tbody> </table>
  
 
</td></tr> </table>
 
</td></tr> </table>
  
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]]]:
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The stable homotopy groups $  \pi _ {m}  ^ {S} (  \mathop{\rm SO} ) $
 +
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>
+
<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"> $  m  \mathop{\rm mod}  8 $
 +
</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"> $  \pi _ {m}  ^ {S} (  \mathop{\rm SO} ) $
 +
</td> <td colname="2" style="background-color:white;" colspan="1"> $  \mathbf Z _ {2} $
 +
</td> <td colname="3" style="background-color:white;" colspan="1"> $  \mathbf Z _ {2} $
 +
</td> <td colname="4" style="background-color:white;" colspan="1">0</td> <td colname="5" style="background-color:white;" colspan="1"> $  \mathbf Z $
 +
</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"> $  \mathbf Z $
 +
</td> </tr> </tbody> </table>
  
 
</td></tr> </table>
 
</td></tr> </table>
  
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]]).
+
The image of the Whitehead homomorphism has been completely calculated (cf. [[#References|[4]]], [[#References|[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|Bernoulli numbers]]).
  
 
====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 {{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>
+
<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 $J(X)$ - I" ''Topology'' , '''2''' (1963) pp. 181–195</TD></TR><TR><TD valign="top">[3b]</TD> <TD valign="top"> J.F. Adams, "On the groups $J(X)$ - II" ''Topology'' , '''3''' (1965) pp. 137–171</TD></TR><TR><TD valign="top">[3c]</TD> <TD valign="top"> J.F. Adams, "On the groups $J(X)$ - III" ''Topology'' , '''3''' (1965) pp. 193–222</TD></TR><TR><TD valign="top">[3d]</TD> <TD valign="top"> J.F. Adams, "On the groups $J(X)$ - 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>
 
 
 
 
  
 
====Comments====
 
====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
+
Given a mapping $  f: X \times Y \rightarrow Z $
 +
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>
+
$$
 +
\Gamma f : X \star Y  \rightarrow  SZ
 +
$$
  
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
+
from the [[Join|join]] $  X \star Y $
 +
of $  X $
 +
and $  Y $
 +
to the [[Suspension|suspension]] $  SZ $
 +
of $  Z $,  
 +
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>
+
$$
 +
f \times  \mathop{\rm id} : X \times Y \times I  \rightarrow  Z \times I,\ \
 +
( x, y, t)  \mapsto  ( f( x, y), t).
 +
$$
  
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]].
+
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 <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
+
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
  
<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>
+
$$
 +
\widetilde \phi  : S  ^ {m} \times S  ^ {q-1}  \rightarrow  S  ^ {q-1} .
 +
$$
  
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" />:
+
Applying the Hopf construction to $  \widetilde \phi  $
 +
gives the mapping $  J \phi $:
  
<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>
+
$$
 +
S  ^ {m+q}  \cong  S  ^ {m} \star S  ^ {q-1}  \mathop \rightarrow \limits ^ { {\Gamma \widetilde \phi  }}  \
 +
S( S  ^ {q-1} )  \cong  S  ^ {q} .
 +
$$
  
 
====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 {{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>
 
<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>

Latest revision as of 20:31, 16 January 2024


$ 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 $J(X)$ - I" Topology , 2 (1963) pp. 181–195
[3b] J.F. Adams, "On the groups $J(X)$ - II" Topology , 3 (1965) pp. 137–171
[3c] J.F. Adams, "On the groups $J(X)$ - III" Topology , 3 (1965) pp. 193–222
[3d] J.F. Adams, "On the groups $J(X)$ - 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
How to Cite This Entry:
Whitehead homomorphism. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Whitehead_homomorphism&oldid=34362
This article was adapted from an original article by A.V. Shokurov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article