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 $s\le \frac{1}{2}$ 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.

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 $f\circ (\alpha \ast \beta )=(f\circ \alpha )\ast (f\circ \beta )$, so $f_{\ast }([\alpha ][\beta ])=f_{\ast }([\alpha ])f_{\ast }([\beta ])$; thus $f_{\ast }$ is a group homomorphism.

For every loop $\alpha$, $\mathrm{id}\circ \alpha =\alpha$ 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 $(f_0{)}_{\ast }([\alpha ])=(f_1{)}_{\ast }([\alpha ])$.

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].