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Difference between revisions of "Whitehead multiplication"

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A multiplication in homotopy groups  $  \pi _ {m} ( X) \times \pi _ {n} ( X) \rightarrow \pi _ {m-} n+ 1 ( X) $,  
+
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} $
+
defined by G.W. Whitehead. In  $  S  ^ {k} $
 
one takes a fixed decomposition into two cells  $  e  ^ {0} $
 
one takes a fixed decomposition into two cells  $  e  ^ {0} $
 
and  $  e  ^ {k} $.  
 
and  $  e  ^ {k} $.  
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$  e  ^ {m} $,  
 
$  e  ^ {m} $,  
 
$  e  ^ {n} $,  
 
$  e  ^ {n} $,  
$  e  ^ {m+} n $.  
+
$  e  ^ {m+n} $.  
 
Therefore the characteristic mapping  $  \phi _ {n,m }  $:
 
Therefore the characteristic mapping  $  \phi _ {n,m }  $:
  
 
$$  
 
$$  
\partial  e  ^ {n+} m =  S  ^ {n+} m- 1  \rightarrow  S  ^ {m} \times S  ^ {n}
+
\partial  e  ^ {n+m}  =  S  ^ {n+ m- 1} \rightarrow  S  ^ {m} \times S  ^ {n}
 
$$
 
$$
  
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$$  
 
$$  
S  ^ {m+} n- 1  \mathop \rightarrow \limits ^ { {W( m,n) }}  S  ^ {m} \lor S  ^ {n}  \rightarrow  S  ^ {m} \times
+
S  ^ {m+ n- 1}   \mathop \rightarrow \limits ^ { {W( m,n) }}  S  ^ {m} \lor S  ^ {n}  \rightarrow  S  ^ {m} \times
 
S  ^ {n} ,
 
S  ^ {n} ,
 
$$
 
$$
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represented by mappings  $  f $
 
represented by mappings  $  f $
 
and  $  g $.  
 
and  $  g $.  
Then the Whitehead product  $  [ \alpha , \beta ] \in \pi _ {n+} m- 1 ( X) $
+
Then the Whitehead product  $  [ \alpha , \beta ] \in \pi _ {n+ m- 1} ( X) $
 
is given by the composition
 
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.
+
S  ^ {m+ n- 1}   \mathop \rightarrow \limits ^ { {W( n,m) }}  S  ^ {m} \lor S  ^ {n}  \mathop \rightarrow \limits ^ { {f\lor g }}  X.
 
$$
 
$$
  
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2) if  $  \alpha , \beta \in \pi _ {1} ( X) $,  
 
2) if  $  \alpha , \beta \in \pi _ {1} ( X) $,  
then  $  [ \alpha , \beta ] = \alpha \beta \alpha  ^ {-} 1 \beta  ^ {-} 1 $;
+
then  $  [ \alpha , \beta ] = \alpha \beta \alpha  ^ {-1} \beta  ^ {-1} $;
  
 
3) if  $  X $
 
3) if  $  X $
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is a generator, is equal to twice the generator of  $  \pi _ {3} ( S  ^ {2} ) $;
 
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 ) $
+
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} ) $,  
+
is generated by one element,  $  [ i _ {2n} , i _ {2n} ] \in \pi _ {4n-1} ( S  ^ {2n} ) $,  
 
where  $  i _ {2n} \in \pi _ {2n} ( S  ^ {2n} ) $
 
where  $  i _ {2n} \in \pi _ {2n} ( S  ^ {2n} ) $
 
is the canonical generator.
 
is the canonical generator.

Latest revision as of 15:48, 29 March 2021


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=51702
This article was adapted from an original article by A.V. Shokurov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article