Whitehead multiplication

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

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