Proposition 1

Let X be a topological space, let $x,y,z,u$ be points of X, and let $\gamma_1\in \varpi_{x,y}(X),\gamma_2\in \varpi_{y,z}(X),\gamma_3\in \varpi_{z,u}(X)$ be path classes. One has

Let $c_1\in \Lambda_{x,y}(X),c_2\in \Lambda_{y,z}(X),c_3\in \Lambda_{z,u}(X)$ be representatives of $\gamma_1,\gamma_2$ and $\gamma_3$ respectively. Let $\varphi :\mathbf{I}\rightarrow \mathbf{I}$ be the function defined by

$t/2$ for $0\leqslant t\leqslant 1/2$,

(4) $\varphi (t) =t-1/4$ for $1/2\leqslant t\leqslant 3/4$,

$2t-1$ for $3/4\leqslant t\leqslant 1$.

The function $\varphi$ is affine on each of the three intervals $[0,1/2]$, $[1/2,3/4]$ and $[3/4,1]$; it is therefore continuous. One has $\varphi (0) = 0$ and $\varphi (1) = 1$. It follows from formula (1) of III, p. 256 defining the juxtaposition of paths that

By the example of III, p. 289, the paths $c_1*(c_2*c_3)$ and $(c_1*c_2)*c_3$ are strictly homotopic, whence equality (3).

Analogously, the function $\psi :\mathbf{I}\rightarrow \mathbf{I}$ defined by

$2t$ for $0\leqslant t\leqslant 1/2$,

1 for $1/2\leqslant t\leqslant 1$

is continuous and satisfies $\psi (0) = 0,\psi (1) = 1$. The equality $\gamma_1\varepsilon_y=\gamma_1$ then follows from the fact that

and the equality $\varepsilon_x\gamma_1=\gamma_1$ is proved analogously, whence (1).

The mapping $\sigma :\mathbf{I}\times \mathbf{I}\rightarrow X$ defined by

$c_1(2ts)$ for $0\leqslant t\leqslant 1/2$,

$c_1(2(1-t)s)$ for $1/2\leqslant t\leqslant 1$

is continuous; it is a strict homotopy joining the path $e_x$ to the path $c_1*\overline{c_1}$, whence the first equality in (2). The second results from the first and from the fact that, for every path $c$, one has $c=\overline{\overline{c}}$.