nLab composition operation

Idea

Composition operations give the action of the homotopy groups of spheres on the homotopy groups of an arbitrary pointed space.

Let $X$ be a pointed space then, of course, its $r^{th}$-homotopy group $\pi_r(X)$ can be defined as the group of pointed homotopy classes of pointed maps from $S^r$ to $X$. In particular $\pi_k(S^r)$ consists of classes of maps from $S^k$ to $S^r$. If $\alpha\in \pi_k(S^r)$ and is represented by $a: S^k \to S^r$, and $\phi\in \pi_r(X)$, represented by $f: S^r\to X$ then the composite $f\circ a: S^k \to X$ represents a class $\phi\cdot \alpha\in \pi_k(X)$.

This composition operation is well defined. It forms one of the primary homotopy operations. An abstraction of this is a component part of the definitional structure of a Pi-algebra.