Loop classes form the group ${\pi }_1(X,x_0)$ under concatenation

If $A$ is an endpoint-fixed homotopy from ${\alpha }_0$ to ${\alpha }_1$ and $B$ one from ${\beta }_0$ to ${\beta }_1$, define $K(s,t)=A(2s,t)$ for $s\le \frac{1}{2}$ and $K(s,t)=B(2s-1,t)$ for $s\ge \frac{1}{2}$; the clauses agree because $A(1,t)=x_0=B(0,t)$, and [L2], [L4], and [L6] make $K$ continuous, so ${\alpha }_0\ast {\beta }_0$ and ${\alpha }_1\ast {\beta }_1$ are endpoint-homotopic. Thus the proposed product is independent of representatives.

If $\varphi :I\to I$ is continuous with $\varphi (0)=0$ and $\varphi (1)=1$, then [L5] makes $(s,t)\mapsto (1-t)s+t\varphi (s)$ continuous. Composing with $\alpha$ by [L4] gives a continuous $H(s,t):=\alpha ((1-t)s+t\varphi (s))$; it takes values in $I$, fixes $s=0,1$, and joins $\alpha$ to $\alpha \circ \varphi$. Hence endpoint-fixing reparametrisation preserves a path class.

The formula $H(s,t)=\alpha (2s(1-t))$ for $s\le \frac{1}{2}$ and $H(s,t)=\alpha (2(1-s)(1-t))$ for $s\ge \frac{1}{2}$ is continuous by [L2], [L4], and [L6], fixes both endpoints at $x_0$, starts at $\alpha \ast \bar{\alpha }$, and ends at $c_{x_0}$; applying the same construction to $\bar{\alpha }$ contracts $\bar{\alpha }\ast \alpha$.

Put $w:=(\alpha \ast \beta )\ast \gamma$ and define $\varphi (s)=s\mathrm{/}2$ for $0\le s\le \frac{1}{2}$, $\varphi (s)=s-\frac{1}{4}$ for $\frac{1}{2}\le s\le \frac{3}{4}$, and $\varphi (s)=2s-1$ for $\frac{3}{4}\le s\le 1$; [L2] makes this continuous, the clauses agree at the seams, $\varphi$ fixes $0,1$, and direct substitution gives $w\circ \varphi =\alpha \ast (\beta \ast \gamma )$, so step 1.2 proves associativity on classes.

For ${\varphi }_R(s)=\min \{2s,1\}$ one has $\alpha \circ {\varphi }_R=\alpha \ast c_{x_0}$, and for ${\varphi }_L(s)=\max \{2s-1,0\}$ one has $\alpha \circ {\varphi }_L=c_{x_0}\ast \alpha$; these continuous endpoint-fixing maps and step 1.2 show that $[c_{x_0}]$ is a two-sided identity.

Step 1.1 gives a well-defined operation, step 2.1 gives associativity, step 2.2 gives the identity, and step 1.3 gives the two-sided inverse $[\bar{\alpha }]$; these are exactly the axioms in [L7], so ${\pi }_1(X,x_0)$ is a group.