The pure braid group $P{B}_n$ as the fundamental group of an ordered configuration space

Fix $n\in \mathbb{N}$ and a base configuration $q=(q_1,\dots ,q_n)$, an ordered $n$-tuple of pairwise distinct points of the open unit disc $\mathrm{int}{D}^2=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }<1\}$, so that $q\in F_n(\mathrm{int}{D}^2)\subseteq F_n(D^2),D^2=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }\le 1\}$ in the notation of The interior-disc and closed-disc configuration spaces are homotopy equivalent and Ordered configuration spaces $F_n(X)$. The pure configuration braid group on $n$ strands is the fundamental group (Based loops and the fundamental group)

$P{B}_n:={\pi }_1(F_n(D^2),q),$

the group of based-loop classes at $q$ in the ordered configuration space of the closed disc, with the first-then-second loop product.

The open-disc model. The inclusion $F_n(\mathrm{int}{D}^2)\to F_n(D^2)$ induces an isomorphism ${\pi }_1(F_n(\mathrm{int}{D}^2),q)⟶{\pi }_1(F_n(D^2),q)$ of fundamental groups at the same base configuration $q$ (The homomorphism on fundamental groups induced by a pointed continuous map); this is claim 3 of The interior-disc and closed-disc configuration spaces are homotopy equivalent. The isomorphism is canonical — it is induced by the inclusion and involves no choice — so on this page and its consumers $P{B}_n$ may be computed from either the closed-disc or the open-disc ordered configuration space. The closed-disc model is the one that matches the geometrically drawn braids, and the open-disc model is the one to which the forgetful fibrations for boundaryless manifolds apply; both give the same group by the displayed isomorphism.

The basepoint. The configuration $q$ is part of the data defining $P{B}_n$, and all groups on this page use the same $q$. Different choices of base configuration give isomorphic groups: $F_n(D^2)$ is path-connected for every $n$ (claim 2 of Ordered configuration spaces cover the unordered ones regularly with deck group $S_n$), and a path between base configurations conjugates loop classes and induces an isomorphism (Conjugating loop classes by a path is an isomorphism of fundamental groups). No particular such isomorphism is fixed here. For $n=0$ the space $F_0(D^2)$ is a point, so $P{B}_0$ is the one-element group; for $n=1$ single-coordinate evaluation gives $F_1(D^2)\cong D^2$ and $P{B}_1$ is trivial.

Scope. This is the configuration-space definition of the pure braid group, stated before any comparison with geometric strands or with the Artin presentation: no presentation of $P{B}_n$ is asserted here. The description by geometric braids and the Artin presentation, and the configuration braid short exact sequence relating $P{B}_n$ to the unordered configuration braid group, belong to the later items on braids, which consume this definition.