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 $$\pi_l(X)$$