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A multiplication in homotopy groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097790/w0977901.png" />, defined by G.W. Whitehead . In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097790/w0977902.png" /> one takes a fixed decomposition into two cells <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097790/w0977903.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097790/w0977904.png" />. Then the product of spheres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097790/w0977905.png" /> has a decomposition into cells <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097790/w0977906.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097790/w0977907.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097790/w0977908.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097790/w0977909.png" />. Therefore the characteristic mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097790/w09779010.png" />:
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<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/w097790/w09779011.png" /></td> </tr></table>
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A multiplication in homotopy groups  $  \pi _ {m} ( X) \times \pi _ {n} ( X) \rightarrow \pi _ {m-} n+ 1 ( X) $,
 +
defined by G.W. Whitehead . In  $  S  ^ {k} $
 +
one takes a fixed decomposition into two cells  $  e  ^ {0} $
 +
and  $  e  ^ {k} $.  
 +
Then the product of spheres  $  S  ^ {m} \times S  ^ {n} $
 +
has a decomposition into cells  $  e  ^ {0} $,
 +
$  e  ^ {m} $,
 +
$  e  ^ {n} $,
 +
$  e  ^ {m+} n $.  
 +
Therefore the characteristic mapping  $  \phi _ {n,m }  $:
 +
 
 +
$$
 +
\partial  e  ^ {n+} m  =  S  ^ {n+} m- 1  \rightarrow  S  ^ {m} \times S  ^ {n}
 +
$$
  
 
factorizes as
 
factorizes as
  
<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/w097790/w09779012.png" /></td> </tr></table>
+
$$
 +
S  ^ {m+} n- 1  \mathop \rightarrow \limits ^ { {W( m,n) }}  S  ^ {m} \lor S  ^ {n}  \rightarrow  S  ^ {m} \times
 +
S  ^ {n} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097790/w09779013.png" /> is a bouquet of spheres. Now, take classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097790/w09779014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097790/w09779015.png" />, represented by mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097790/w09779016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097790/w09779017.png" />. Then the Whitehead product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097790/w09779018.png" /> is given by the composition
+
where $  S  ^ {m} \lor S  ^ {n} $
 +
is a bouquet of spheres. Now, take classes $  \alpha \in \pi _ {m} ( X) $
 +
and $  \beta \in \pi _ {n} ( X) $,  
 +
represented by mappings $  f $
 +
and $  g $.  
 +
Then the Whitehead product $  [ \alpha , \beta ] \in \pi _ {n+} m- 1 ( X) $
 +
is given by the composition
  
<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/w097790/w09779019.png" /></td> </tr></table>
+
$$
 +
S  ^ {m+} n- 1  \mathop \rightarrow \limits ^ { {W( n,m) }}  S  ^ {m} \lor S  ^ {n}  \mathop \rightarrow \limits ^ { {f\lor g }}  X.
 +
$$
  
 
The following properties are satisfied by this product:
 
The following properties are satisfied by this product:
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097790/w09779020.png" />;
+
1) $  [ \alpha , \beta ] = (- 1) ^ { \mathop{\rm deg}  \alpha  \mathop{\rm deg}  \beta } [ \beta , \alpha ] $;
  
2) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097790/w09779021.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097790/w09779022.png" />;
+
2) if $  \alpha , \beta \in \pi _ {1} ( X) $,  
 +
then $  [ \alpha , \beta ] = \alpha \beta \alpha  ^ {-} 1 \beta  ^ {-} 1 $;
  
3) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097790/w09779023.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097790/w09779024.png" />-simple, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097790/w09779025.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097790/w09779026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097790/w09779027.png" />;
+
3) if $  X $
 +
is $  n $-
 +
simple, then $  [ \alpha , \beta ] = 0 $
 +
for $  \alpha \in \pi _ {1} ( X) $,  
 +
$  \beta \in \pi _ {n} ( X) $;
  
4) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097790/w09779028.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097790/w09779029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097790/w09779030.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097790/w09779031.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097790/w09779032.png" />-simple;
+
4) if $  [ \alpha , \beta ]= 0 $
 +
for all $  \alpha \in \pi _ {1} ( X) $,  
 +
$  \beta \in \pi _ {n} ( X) $,  
 +
then $  X $
 +
is $  n $-
 +
simple;
  
5) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097790/w09779033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097790/w09779034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097790/w09779035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097790/w09779036.png" />, then
+
5) if $  \alpha \in \pi _ {n} ( X) $,
 +
$  \beta \in \pi _ {m} ( X) $,
 +
$  \gamma \in \pi _ {k} ( X) $,  
 +
$  n , m, k > 1 $,  
 +
then
  
<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/w097790/w09779037.png" /></td> </tr></table>
+
$$
 +
(- 1)  ^ {nk} [[ \alpha , \beta ] , \gamma ] +(- 1)  ^ {mn} [[ \beta , \gamma ] ,\
 +
\alpha ] + (- 1)  ^ {mk} [[ \gamma , \alpha ] , \beta ]  = 0;
 +
$$
  
6) the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097790/w09779038.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097790/w09779039.png" /> is a generator, is equal to twice the generator of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097790/w09779040.png" />;
+
6) the element $  [ i _ {1} , i _ {2} ] \in \pi _ {3} ( S  ^ {2} ) $,  
 +
where $  i _ {2} \in \pi _ {2} ( S  ^ {2} )= \mathbf Z $
 +
is a generator, is equal to twice the generator of $  \pi _ {3} ( S  ^ {2} ) $;
  
7) the kernel of the epimorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097790/w09779041.png" /> is generated by one element, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097790/w09779042.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097790/w09779043.png" /> is the canonical generator.
+
7) the kernel of the epimorphism $  \Sigma : \pi _ {4n-} 1 ( S  ^ {2n} ) \rightarrow \pi _ {4n} ( S  ^ {2n+} 1 ) $
 +
is generated by one element, $  [ i _ {2n} , i _ {2n} ] \in \pi _ {4n-} 1 ( S  ^ {2n} ) $,  
 +
where $  i _ {2n} \in \pi _ {2n} ( S  ^ {2n} ) $
 +
is the canonical generator.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1a]</TD> <TD valign="top">  G.W. Whitehead,  "On products in homotopy groups"  ''Ann. of Math.'' , '''47'''  (1946)  pp. 460–475</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></table>
 
<table><TR><TD valign="top">[1a]</TD> <TD valign="top">  G.W. Whitehead,  "On products in homotopy groups"  ''Ann. of Math.'' , '''47'''  (1946)  pp. 460–475</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></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.W. Whitehead,  "Elements of homotopy theory" , Springer  (1978)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.H. Spanier,  "Algebraic topology" , McGraw-Hill  (1966)  pp. 419–420</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S.-T. Hu,  "Homotopy theory" , Acad. Press  (1959)  pp. 138–139</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.W. Whitehead,  "Elements of homotopy theory" , Springer  (1978)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.H. Spanier,  "Algebraic topology" , McGraw-Hill  (1966)  pp. 419–420</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S.-T. Hu,  "Homotopy theory" , Acad. Press  (1959)  pp. 138–139</TD></TR></table>

Revision as of 08:29, 6 June 2020


A multiplication in homotopy groups $ \pi _ {m} ( X) \times \pi _ {n} ( X) \rightarrow \pi _ {m-} n+ 1 ( X) $, defined by G.W. Whitehead . In $ S ^ {k} $ one takes a fixed decomposition into two cells $ e ^ {0} $ and $ e ^ {k} $. Then the product of spheres $ S ^ {m} \times S ^ {n} $ has a decomposition into cells $ e ^ {0} $, $ e ^ {m} $, $ e ^ {n} $, $ e ^ {m+} n $. Therefore the characteristic mapping $ \phi _ {n,m } $:

$$ \partial e ^ {n+} m = S ^ {n+} m- 1 \rightarrow S ^ {m} \times S ^ {n} $$

factorizes as

$$ S ^ {m+} n- 1 \mathop \rightarrow \limits ^ { {W( m,n) }} S ^ {m} \lor S ^ {n} \rightarrow S ^ {m} \times S ^ {n} , $$

where $ S ^ {m} \lor S ^ {n} $ is a bouquet of spheres. Now, take classes $ \alpha \in \pi _ {m} ( X) $ and $ \beta \in \pi _ {n} ( X) $, represented by mappings $ f $ and $ g $. Then the Whitehead product $ [ \alpha , \beta ] \in \pi _ {n+} m- 1 ( X) $ is given by the composition

$$ S ^ {m+} n- 1 \mathop \rightarrow \limits ^ { {W( n,m) }} S ^ {m} \lor S ^ {n} \mathop \rightarrow \limits ^ { {f\lor g }} X. $$

The following properties are satisfied by this product:

1) $ [ \alpha , \beta ] = (- 1) ^ { \mathop{\rm deg} \alpha \mathop{\rm deg} \beta } [ \beta , \alpha ] $;

2) if $ \alpha , \beta \in \pi _ {1} ( X) $, then $ [ \alpha , \beta ] = \alpha \beta \alpha ^ {-} 1 \beta ^ {-} 1 $;

3) if $ X $ is $ n $- simple, then $ [ \alpha , \beta ] = 0 $ for $ \alpha \in \pi _ {1} ( X) $, $ \beta \in \pi _ {n} ( X) $;

4) if $ [ \alpha , \beta ]= 0 $ for all $ \alpha \in \pi _ {1} ( X) $, $ \beta \in \pi _ {n} ( X) $, then $ X $ is $ n $- simple;

5) if $ \alpha \in \pi _ {n} ( X) $, $ \beta \in \pi _ {m} ( X) $, $ \gamma \in \pi _ {k} ( X) $, $ n , m, k > 1 $, then

$$ (- 1) ^ {nk} [[ \alpha , \beta ] , \gamma ] +(- 1) ^ {mn} [[ \beta , \gamma ] ,\ \alpha ] + (- 1) ^ {mk} [[ \gamma , \alpha ] , \beta ] = 0; $$

6) the element $ [ i _ {1} , i _ {2} ] \in \pi _ {3} ( S ^ {2} ) $, where $ i _ {2} \in \pi _ {2} ( S ^ {2} )= \mathbf Z $ is a generator, is equal to twice the generator of $ \pi _ {3} ( S ^ {2} ) $;

7) the kernel of the epimorphism $ \Sigma : \pi _ {4n-} 1 ( S ^ {2n} ) \rightarrow \pi _ {4n} ( S ^ {2n+} 1 ) $ is generated by one element, $ [ i _ {2n} , i _ {2n} ] \in \pi _ {4n-} 1 ( S ^ {2n} ) $, where $ i _ {2n} \in \pi _ {2n} ( S ^ {2n} ) $ is the canonical generator.

References

[1a] G.W. Whitehead, "On products in homotopy groups" Ann. of Math. , 47 (1946) pp. 460–475
[1b] G.W. Whitehead, "A generalization of the Hopf invariant" Ann. of Math. , 51 (1950) pp. 192–237

Comments

References

[a1] G.W. Whitehead, "Elements of homotopy theory" , Springer (1978)
[a2] E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) pp. 419–420
[a3] S.-T. Hu, "Homotopy theory" , Acad. Press (1959) pp. 138–139
How to Cite This Entry:
Whitehead multiplication. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Whitehead_multiplication&oldid=18732
This article was adapted from an original article by A.V. Shokurov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article