Whitehead product

This article describes a binary operation involving the homotopy groups of a topological space

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