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