Toda bracket

Definition

Let $W$, $X$, $Y$, $Z$ be pointed spaces, and let $f \colon W \to X$, $g \colon X \to Y$, $h \colon Y \to Z$ be pointed maps such that $g \circ f$ and $h \circ g$ are null-homotopic. Let $CW = (W \times [0, 1]) / (W \times \set{1} \cup \set{w_0} \times [0, 1])$ be the reduced cone on $W$, with $W$ included as $W \times \set{0}$.

1. Choose a pointed null-homotopy of $g \circ f$, that is, a map $A \colon CW \to Y$ that restricts to $g \circ f$ on $W$.
2. Choose a pointed null-homotopy of $h \circ g$, that is, a map $B \colon CX \to Z$ that restricts to $h \circ g$ on $X$.
3. The maps $h \circ A \colon CW \to Z$ and $B \circ Cf \colon CW \to Z$ both restrict to $h \circ g \circ f$ on $W$. Gluing them along $W$ gives a map from two cones on $W$ glued along their bases, which is the reduced suspension $\Sigma W$, to $Z$.

The Toda bracket $\langle h, g, f \rangle$ is the set of homotopy classes of all maps $\Sigma W \to Z$ obtained in this way, over all choices of $A$ and $B$.