Composite of homotopies

Definition

Definition assuming all homotopies take time $1$

Suppose:

• $f_1,f_2,f_3:X\to Y$ are continuous maps
• $I$ is the closed unit interval
• $F_{12}:X\times I\to Y$ is a homotopy from $f_1$ to $f_2$
• $F_{23}:X\times I\to Y$ is a homotopy from $f_2$ to $f_3$

Then, the composite of homotopies $F_{13}=F_{12}*F_{23}$ is a homotopy from $f_1$ to $f_3$ given as follows:

$F_{13}(x,t)=\{\begin{array}{cc}{F}_{12}(x,2t),& 0\le t<1/2\\ F_{23}(x,2t-1),& 1/2\le t\le 1\end{array}$