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

${\pi }_1(X,x_0):=\{[\alpha ]:\alpha \text{ is a based loop at }{x}_0\}\mathrm{.}$

For composable paths, write

$(\alpha \ast \beta )(s):=\{\begin{array}{ll}\alpha (2s),& 0\le s\le \frac{1}{2},\\ \beta (2s-1),& \frac{1}{2}\le s\le 1.\end{array}$

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

$[\alpha ][\beta ]:=[\alpha \ast \beta ]\mathrm{.}$

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.