Based loops and the fundamental group

Let $X$ be a topological space and let $x_0\in X$. A based loop at $x_0$ is a path $\alpha:I\to X$ with $\alpha(0)=x_0=\alpha(1)$ (Paths, path-connected spaces and path components). Two based loops are equivalent when they are path-homotopic relative to the endpoints. This is an equivalence relation by Homotopy relative to a fixed subspace, and path homotopy relative to endpoints, are equivalence relations.

The fundamental group set of $X$ at $x_0$ is

$\pi_1(X,x_0):=\{\,[\alpha]:\alpha\text{ is a based loop at }x_0\,\}.$

For composable paths, write

$(\alpha*\beta)(s):=\begin{cases}\alpha(2s),&0\leq s\leq\tfrac12,\\ \beta(2s-1),&\tfrac12\leq s\leq1.\end{cases}$

The finite closed-pasting argument already carried out in Paths, path-connected spaces and path components shows that this is a path. The multiplication proposed on loop classes is

$[\alpha][\beta]:=[\alpha*\beta].$

Order convention. The product $[\alpha][\beta]$ traverses $\alpha$ first and $\beta$ second. Every product on this page uses this convention.

The constant loop at $x_0$ is denoted $c_{x_0}$, and the reversed loop is $\bar\alpha(s):=\alpha(1-s)$. The next theorem proves that multiplication is independent of representatives and that $[c_{x_0}]$ and $[\bar\alpha]$ are the identity and inverse required by the group axioms.