Whitehead product

Definition

The Whitehead product is a product relating the homotopy groups of a topological space as follows:

$\pi _{k}(X)\times \pi _{l}(X)\to \pi _{k+l-1}(X)$

It is defined as follows: consider the CW-complex structure on $S^{k}\times S^{l}$. The gluing of the $k+l$-cell is achieved by an attaching map from $S^{k+l-1}$ to $S^{k}\vee S^{l}$, and hence we can get a based map from $S^{k+l-1}$ to $X$ using based maps from $S^{k}$ to $X$ and $S^{l}$ to $X$. This is designated as the Whitehead product.

Facts

• The Whitehead product when $k=l=1$ returns the commutator of the two elements
• The Whitehead product when $k=1$, is $(g,a)\mapsto g.a-a$ where $g.a$ arises from the natural action of $\pi _{1}(X)$ on $\pi _{l}(X)$ Further information: Actions of the fundamental group