Proposition 6

Let X be an arcwise connected topological space and let $x$ be a point of X.

a) Every loop in X is freely homotopic to a loop at $x$. More precisely, if $c$ is a loop at $y$ and $d$ a path with origin $y$ and end $x,c$ is freely homotopic to the loop $(\overline{d}*c)*d$ at $x$.

b) Two loops in X at $x$ are freely homotopic if and only if their strict homotopy classes are conjugate in the group $\pi_1(X, x)$.

Let us prove a). For every $s\in [0,1]$, let $d_s$ denote the path in X defined by $d_s(t) =d(st)$ for $t\in \mathbf{I}$; its origin is $y$. Since the mapping $(s, t)\mapsto d(st)$ is continuous, the mapping $s\mapsto d_s$ of $\mathbf{I}$ into $\mathscr{C}_c(\mathbf{I}; X)$ is continuous (III, p. 257, prop. 1). The mapping $s\mapsto (\overline{d_s}*c)*d_s$ is then a path in Ω(X) (III, p. 257, prop. 2) joining $(e_y*c)*e_y$ to $(\overline{d}*c)*d$, whence a).

Let $c$ and $c'$ be two loops in X at $x$. If their strict homotopy classes are conjugate in $\pi_1(X, x)$, there exists a loop $d$ at $x$ such that $c'$ is strictly homotopic to the loop $(\overline{d}*c)*d$. It follows from a) that $c$ and $c'$ are freely homotopic. Conversely, suppose that there exists a free homotopy $\varphi$ joining $c$ to $c'$. Put $d(t) =\varphi (0, t)$; one has also $d(t) =\varphi (1, t)$ and $d$ is a loop at $x$. By lemma 1 of III, p. 295, the loops $c*d$ and $d*c'$ are strictly homotopic. The strict homotopy classes of $c$ and $c'$ are therefore conjugate in $\pi_1(X, x)$.