View source for Whitehead multiplication

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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

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

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====
<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====

====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>