The Fundamental Group

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 $s\leq\tfrac12$ and $K(s,t)=B(2s-1,t)$ for $s\geq\tfrac12$; the clauses agree because $A(1,t)=x_0=B(0,t)$, and [L2], [L4], and [L6] make $K$ continuous, so $\alpha_0*\beta_0$ and $\alpha_1*\beta_1$ are endpoint-homotopic. Thus the proposed product is independent of representatives.

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

The formula $H(s,t)=\alpha(2s(1-t))$ for $s\leq\tfrac12$ and $H(s,t)=\alpha(2(1-s)(1-t))$ for $s\geq\tfrac12$ is continuous by [L2], [L4], and [L6], fixes both endpoints at $x_0$, starts at $\alpha*\bar\alpha$, and ends at $c_{x_0}$; applying the same construction to $\bar\alpha$ contracts $\bar\alpha*\alpha$.

Put $w:=(\alpha*\beta)*\gamma$ and define $\phi(s)=s/2$ for $0\leq s\leq\tfrac12$, $\phi(s)=s-\tfrac14$ for $\tfrac12\leq s\leq\tfrac34$, and $\phi(s)=2s-1$ for $\tfrac34\leq s\leq1$; [L2] makes this continuous, the clauses agree at the seams, $\phi$ fixes $0,1$, and direct substitution gives $w\circ\phi=\alpha*(\beta*\gamma)$, so step 1.2 proves associativity on classes.

For $\phi_R(s)=\min\{2s,1\}$ one has $\alpha\circ\phi_R=\alpha*c_{x_0}$, and for $\phi_L(s)=\max\{2s-1,0\}$ one has $\alpha\circ\phi_L=c_{x_0}*\alpha$; 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.

If $\alpha_0$ and $\alpha_1$ are endpoint-homotopic, postcomposing their homotopy by $f$ gives an endpoint-homotopy from $f\circ\alpha_0$ to $f\circ\alpha_1$ by [L2]; hence the formula in [L1] is independent of the representative.

The concatenation formulas give the literal equality $f\circ(\alpha*\beta)=(f\circ\alpha)*(f\circ\beta)$, so $f_*([\alpha][\beta])=f_*([\alpha])f_*([\beta])$; thus $f_*$ is a group homomorphism.

For every loop $\alpha$, $\operatorname{id}\circ\alpha=\alpha$ and $(g\circ f)\circ\alpha=g\circ(f\circ\alpha)$, which proves the identity and composition formulas on every loop class.

If $H$ is a homotopy from $f_0$ to $f_1$ fixing $x_0$, precomposition by a based loop $\alpha$ gives an endpoint-fixed path homotopy from $f_0\circ\alpha$ to $f_1\circ\alpha$ by [L2], so $(f_0)_*([\alpha])=(f_1)_*([\alpha])$.

Steps 1.1--1.4 prove well-definedness, the homomorphism law, functoriality and based-homotopy invariance.

Apply [L1] to the maps $\alpha:I\to C$ and $c_{x_0}:I\to C$; it gives the displayed continuous homotopy $H(s,t)=(1-t)\alpha(s)+t x_0$.

Since $\alpha(0)=\alpha(1)=x_0$, one has $H(0,t)=x_0=H(1,t)$ for every $t$, so this homotopy is relative to the endpoints.

Steps 1.1 and 2.1 show that every loop at every basepoint represents the constant-loop class, so each fundamental group has one element; [L2] supplies nonempty path-connectedness.

Therefore $C$ is simply connected by [L4].