Based motions of an unordered point configuration

Fix $n\in \mathbb{N}$ and let $Q=(q_1,\dots ,q_n)$ be the explicit base tuple fixed in Geometric braids in the disc with setwise endpoints. Identify its real closed unit disk with the complex closed unit disk $D^2$ by $(x,y)\mapsto x+iy$, so $Q\in F_n(\mathrm{int}{D}^2)$ and its orbit $[Q]\in C_n(D^2)$ is defined by Ordered configuration spaces $F_n(X)$ and Unordered configuration spaces $C_n(X)$.

An unordered configuration motion is a continuous map $\alpha :I⟶C_n(D^2)\mathrm{.}$ It is a based motion at $Q$ if $\alpha (0)=[Q]=\alpha (1),$ and it is an interior based motion if, in addition, $\alpha (t)\in C_n(\mathrm{int}{D}^2)\text{for every }t\in I\mathrm{.}$ Every slice is a collision-free $n$-point set because it is an element of the unordered configuration space; continuity is with respect to its quotient topology. With the base configuration in The configuration braid group $B_n^{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{f}}$ as the fundamental group of an unordered configuration space chosen to be this same $Q$, the based-loop classes of all such closed-disc motions form $B_n^{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{f}}={\pi }_1(C_n(D^2),[Q])\mathrm{.}$

Let $\iota :C_n(\mathrm{int}{D}^2)\hookrightarrow C_n(D^2)$ be the inclusion. The published radial homotopy in The interior-disc and closed-disc configuration spaces are homotopy equivalent shows that ${\iota }_{\ast }$ is an isomorphism at the same basepoint $[Q]$. Thus every class in $B_n^{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{f}}$ has an interior based-motion representative, and two interior motions represent the same closed-disc class exactly when they are path-homotopic through interior based motions. This transfers classes between the two disc models; it does not assert that every closed-disc motion itself stays in the interior.

For $n=0$, both configuration spaces are one-point spaces and there is one based motion. For $n=1$, $C_1(X)$ is canonically homeomorphic to $X$ and the geometric basepoint is $q_1=0$, so the same definitions apply without any collision condition. No choice principle is used: the base tuple is the fixed one from the geometric-braid definition, and the quotient and inclusion maps are specified maps.