Toda bracket

Summary

The Toda bracket is a secondary composition operation on homotopy classes of maps. It takes three maps $f$, $g$, $h$ such that $g \circ f$ and $h \circ g$ are null-homotopic. It produces a set of maps out of the suspension of the source of $f$. Toda introduced it to compute the homotopy groups of spheres. It remains a basic tool for finding relations in the stable homotopy groups of spheres.