nLab
composition operation

The 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 $r^{\mathrm{th}}$-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 $\varphi \in {\pi }_r(X)$, represented by $f:S^r\to X$ then the composite $f\circ a:S^k\to X$ so represents a class $\varphi \cdot \alpha \in {\pi }_k(X)$.

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.