Proposition 3

Let X and Y be topological spaces, $f_0$ and $f_1$ continuous mappings of X into Y, and let $\sigma : X\times \mathbf{I}\rightarrow$ Y be a homotopy joining $f_0$ to $f_1$. Let $x$ be a point of X; put $y_0=f_0(x)$, $y_1=f_1(x)$, and denote by $\delta \in \varpi_{y_0,y_1}(Y)$ the class of the path $d$ defined by $d(t) =\sigma (x, t)$ for $t\in \mathbf{I}$. For every $\gamma \in \pi_1(X, x)$, one has $(f_1)_*(\gamma ) =$ $\delta^{-1}((f_0)_*(\gamma ))\delta$.

Let $c$ be a loop in X at $x$ and let $\gamma$ be its strict homotopy class. Put $\gamma_0= (f_0)_*(\gamma )$ and $\gamma_1= (f_1)_*(\gamma )$; these are the strict homotopy classes of $f_0\circ c$ and $f_1\circ c$. For $(t, s)\in \mathbf{I}\times \mathbf{I}$, put $\varphi (t, s) =\sigma (c(t), s)$. For every $t\in \mathbf{I}$, one has $\varphi (t,0) = (f_0\circ c)(t),\varphi (t,1) = (f_1\circ c)(t)$ and $\varphi (0, t) =\varphi (1, t) =d(t)$. The relation $\gamma_0\delta =\delta \gamma_1$ therefore follows from the following lemma.