Free homotopy classes of loops are conjugacy classes

The square homotopy that realizes the conjugation. Assume a free homotopy $H$ from $\alpha$ to $\beta$ is given and let $\gamma (t):=H(0,t)=H(1,t)$; then $\gamma$ is a loop at $x_0$, because $\gamma (0)=H(0,0)=\alpha (0)=x_0$ and $\gamma (1)=H(0,1)=\beta (0)=x_0$. Define the piecewise-affine path in the square $B\text{\hspace{0.17em}⁣}:I\to I\times I$ by $B(s):=(0,1-4s)$ for $s\le \frac{1}{4}$, $B(s):=(4s-1,0)$ for $\frac{1}{4}\le s\le \frac{1}{2}$, and $B(s):=(1,2s-1)$ for $s\ge \frac{1}{2}$, and define $\phi (s,t):=(1-t)(s,1)+tB(s)\in I\times I$ and $J:=H\circ \phi \text{\hspace{0.17em}⁣}:I\times I\to X$. For joint continuity on a rectangle, the product topology has neighborhoods $\mathrm{\mid }s-s_0\mathrm{\mid }<\delta$, $\mathrm{\mid }t-t_0\mathrm{\mid }<\delta$. At a point $(a_0,b_0)$, $\mathrm{\mid }(a+b)-(a_0+b_0)\mathrm{\mid }\le \mathrm{\mid }a-a_0\mathrm{\mid }+\mathrm{\mid }b-b_0\mathrm{\mid }$ and $\mathrm{\mid }ab-a_0{b}_0\mathrm{\mid }\le \mathrm{\mid }a\mathrm{\mid }\mathrm{\mid }b-b_0\mathrm{\mid }+\mathrm{\mid }{b}_0\mathrm{\mid }\mathrm{\mid }a-a_0\mathrm{\mid }$, with $\mathrm{\mid }a\mathrm{\mid }\le \mathrm{\mid }{a}_0\mathrm{\mid }+1$ when $\mathrm{\mid }a-a_0\mathrm{\mid }<1$. These estimates prove joint continuity of addition and multiplication; composing them with continuous coordinates gives continuity of each polynomial expression used below. Every piece of $B$ is affine, so $B$ is continuous by [F5] and [F6]; on each of the three closed pieces every component of $\phi$ is a sum of products of affine coordinate functions, hence is continuous by these estimates; the pieces agree on their seams, so $\phi$ is continuous by [F5], [F6] and [F7]; hence $J$ is continuous by [F5].

The moving-basepoint homotopy. Assume now that $\gamma$ is an arbitrary loop at $x_0$ and define, for $(s,u)\in I\times I$, $R_u(s):=\gamma (u(1-4s))$ for $s\le \frac{1}{4}$, $R_u(s):=\alpha (4s-1)$ for $\frac{1}{4}\le s\le \frac{1}{2}$, and $R_u(s):=\gamma (u(2s-1))$ for $s\ge \frac{1}{2}$. Each $R_u$ is a loop at $\gamma (u)$, because the three pieces join continuously at $s=\frac{1}{4}$ and $s=\frac{1}{2}$, they take the values $\gamma (u),x_0,x_0,\gamma (u)$ at $s=0,\frac{1}{4},\frac{1}{2},1$, and each argument supplied to $\gamma$ or $\alpha$ is a polynomial in $(s,u)$ on its closed rectangle. These arguments are continuous by the estimates in step 1.1, and composing with the continuous loops is permitted by [F5]; consequently the map $R\text{\hspace{0.17em}⁣}:I\times I\to X$, $R(s,u):=R_u(s)$, is continuous by [F5], [F6] and [F7].

The square homotopy is a path homotopy from $\beta$ to $(\bar{\gamma }\ast \alpha )\ast \gamma$. For $J$ of step 1.1: $J(s,0)=H(\phi (s,0))=H(s,1)=\beta (s)$; $J(s,1)=H(B(s))$ and the three pieces of $B$ give $H(B(s))=\gamma (1-4s)=\bar{\gamma }(4s)$ for $s\le \frac{1}{4}$, $H(B(s))=\alpha (4s-1)$ for $\frac{1}{4}\le s\le \frac{1}{2}$ and $H(B(s))=\gamma (2s-1)$ for $s\ge \frac{1}{2}$, which is exactly the loop $(\bar{\gamma }\ast \alpha )\ast \gamma$ of [F1]; finally $\phi (0,t)=(1-t)(0,1)+t(0,1)=(0,1)$ and $\phi (1,t)=(1-t)(1,1)+t(1,1)=(1,1)$, so $J(0,t)=H(0,1)=\beta (0)=x_0$ and $J(1,t)=H(1,1)=\beta (1)=x_0$ for every $t$. Hence $J$ is a path homotopy relative to the endpoints from $\beta$ to $(\bar{\gamma }\ast \alpha )\ast \gamma$ in the sense of [F4].

The moving-basepoint family is a free homotopy. For the family $R$ of step 2.1 one has $R(s,0)=c_{x_0}$-insertions: $R_0(s)=x_0$ for $s\le \frac{1}{4}$ and $s\ge \frac{1}{2}$, while $R_0(s)=\alpha (4s-1)$ in between, so $R_0={\bar{c}}_{x_0}\ast \alpha \ast c_{x_0}$ is the loop obtained from $\alpha$ by adjoining constant loops; and $R_1=\bar{\gamma }\ast \alpha \ast \gamma$. Since $R(0,u)=\gamma (u)=R(1,u)$ for every $u$, the family $R$ is a free homotopy from $R_0$ to $R_1=\bar{\gamma }\ast \alpha \ast \gamma$ in the sense of the Statement.

Free homotopy implies conjugacy. By step 2.2 and [F1] the classes satisfy $[\beta ]=[(\bar{\gamma }\ast \alpha )\ast \gamma ]=[\bar{\gamma }][\alpha ][\gamma ]=[\gamma {]}^{-1}[\alpha ][\gamma ]$, using associativity of the group product and $[\bar{\gamma }]=[\gamma {]}^{-1}$ from [F2].

Conjugacy implies free homotopy. Conversely, let $\gamma$ be a loop at $x_0$ with $[\beta ]=[\gamma {]}^{-1}[\alpha ][\gamma ]$. By [F2] the class of $R_0={\bar{c}}_{x_0}\ast \alpha \ast c_{x_0}$ is $[c_{x_0}{]}^{-1}[\alpha ][c_{x_0}]=[\alpha ]$, so by [F1] there is a path homotopy relative to the endpoints from $\alpha$ to $R_0$; by step 3.1 the family $R$ is a free homotopy from $R_0$ to $\bar{\gamma }\ast \alpha \ast \gamma$; and $[\bar{\gamma }\ast \alpha \ast \gamma ]=[\gamma {]}^{-1}[\alpha ][\gamma ]=[\beta ]$, so again by [F1] there is a path homotopy relative to the endpoints from $\bar{\gamma }\ast \alpha \ast \gamma$ to $\beta$. Reparametrising the homotopy parameter by the three-part affine map $t\mapsto 3t$, $t\mapsto 3t-1$, $t\mapsto 3t-2$ on $[0,\frac{1}{3}],[\frac{1}{3},\frac{2}{3}],[\frac{2}{3},1]$ and pasting the three homotopies, which agree on the seams, gives one continuous $K\text{\hspace{0.17em}⁣}:I\times I\to X$ with $K(s,0)=\alpha (s)$, $K(s,1)=\beta (s)$ and $K(0,t)=K(1,t)$ for every $t$: the pasting is licensed by [F5] and the affine reparametrisation by [F6]. So $\alpha$ and $\beta$ are freely homotopic.

Conclusion. Step 3.2 shows that a free homotopy from $\alpha$ to $\beta$ produces a conjugating loop $\gamma$ with $[\beta ]=[\gamma {]}^{-1}[\alpha ][\gamma ]$, and step 4.1 shows conversely that each conjugating relation produces a free homotopy. Hence $\alpha$ and $\beta$ are freely homotopic if and only if their classes are conjugate. Moreover free homotopy is an equivalence relation: it is reflexive via the constant homotopy $H(s,t):=\alpha (s)$, symmetric by reversing the deformation parameter, and transitive by the pasting argument of step 4.1. Therefore the free homotopy classes of loops at $x_0$ are in bijection with the conjugacy classes of ${\pi }_1(X,x_0)$, the map being induced by $\alpha \mapsto [\alpha ]$.