Smooth representatives of configuration loops

The ordered lift and its margin. Let $p:F_n(\mathrm{int}{D}^2)\to C_n(\mathrm{int}{D}^2)$ be the quotient covering map of [L3]. By [L4] there is a unique path $u:I\to F_n(\mathrm{int}{D}^2)$ with $u(0)=Q_n$ and $p\circ u=\alpha$; its coordinates $u_1,\dots ,u_n$ are continuous and satisfy $u_i(t)\mathrm{\ne }{u}_j(t)$ for $i\mathrm{\ne }j$ and $u_i(t)\in \mathrm{int}{D}^2$. Compactness and finiteness give a positive boundary margin $d:={\min }_{i,t}(1-\parallel u_i(t){\parallel }_2)>0$; if $n\ge 2$, also put $c:={\min }_{i<j,t}\parallel u_i(t)-u_j(t){\parallel }_2>0$, and if $n=1$ put $c:=1$. Then $m:=\min (c,d)>0$ bounds every pairwise separation and every boundary margin from below (with the pairwise condition vacuous for $n=1$).

Smooth approximation with fixed endpoints and flat time ends. Write $u_i=(u_{i,1},u_{i,2})$. For each of the finitely many functions $u_{i,k}$ apply [L1] with ${\epsilon }_1>0$ to obtain a polynomial $P_{i,k}$ with ${\sup }_t\mathrm{\mid }{P}_{i,k}(t)-u_{i,k}(t)\mathrm{\mid }<{\epsilon }_1$, and put $R_{i,k}(t):=P_{i,k}(t)+(1-t)(u_{i,k}(0)-P_{i,k}(0))+t(u_{i,k}(1)-P_{i,k}(1))$, so that $R_{i,k}(0)=u_{i,k}(0)$, $R_{i,k}(1)=u_{i,k}(1)$ and ${\sup }_t\mathrm{\mid }{R}_{i,k}(t)-u_{i,k}(t)\mathrm{\mid }<2{\epsilon }_1$. Now choose a small $\epsilon \in (0,\frac{1}{4})$ and use [L2] to define the smooth time change $\lambda (t):=\sigma ((t-\epsilon )\mathrm{/}\epsilon )\sigma ((1-\epsilon -t)\mathrm{/}\epsilon )t+(1-\sigma ((1-\epsilon -t)\mathrm{/}\epsilon \left)\right)$; it is smooth, equals $0$ on $[0,\epsilon ]$, equals $1$ on $[1-\epsilon ,1]$, satisfies $0\le \lambda (t)\le 1$ and $\mathrm{\mid }\lambda (t)-t\mathrm{\mid }\le 4\epsilon$ on $I$, and equals $t$ on $[2\epsilon ,1-2\epsilon ]$. Set $r_i(t):=(R_{i,1}(t),R_{i,2}(t))$ and $z_i:=r_i\circ \lambda$. Then each $z_i$ is smooth, is constant on $[0,\epsilon ]$ and on $[1-\epsilon ,1]$, and because the finitely many $u_i$ are uniformly continuous there is a modulus of continuity $\omega$ for all of them on $I$ with $\parallel u_i(\lambda (t))-u_i(t){\parallel }_2\le \omega (4\epsilon )$. Choosing ${\epsilon }_1$ and $\epsilon$ so small that $3{\epsilon }_1+\omega (4\epsilon )<m\mathrm{/}4$, we obtain $\parallel z_i(t)-u_i(t){\parallel }_2\le \parallel R_i(\lambda (t))-u_i(\lambda (t)){\parallel }_2+\parallel u_i(\lambda (t))-u_i(t){\parallel }_2<m\mathrm{/}4$ for every $i$ and $t$.

The approximating tuple is collision-free, interior, and based. For $i\mathrm{\ne }j$ and all $t\in [0,1]$ the estimates of step 1.2 give $\parallel z_i(t)-z_j(t){\parallel }_2\ge m-2(m\mathrm{/}4)=m\mathrm{/}2>0$ and $1-\parallel z_i(t){\parallel }_2\ge m-m\mathrm{/}4>0$, so all $z_i(t)$ lie in $\mathrm{int}{D}^2$ and are pairwise distinct. Moreover $z_i(0)=r_i(\lambda (0))=r_i(0)=u_i(0)=q_i$ and $z_i(1)=r_i(1)=u_i(1)$, so $[z(1)]=[u(1)]=\alpha (1)=[Q_n]$: the terminal tuple $z(1)$ is a permutation of $Q_n$, and $z:I\to F_n(\mathrm{int}{D}^2)$ is an ordered path from $Q_n$ to that permutation, while $p\circ z$ is a based loop at $[Q_n]$.

A relative homotopy to the smooth representative. For $s,t\in [0,1]$ put $\gamma (s,t):=(1-s)u(t)+sz(t)$, computed coordinatewise in ${\mathbb{R}}^{2n}$. The map $\gamma$ is continuous, and by the estimates of steps 1.2 and 2.1 every $\gamma (s,t)$ again has pairwise distinct coordinates at distance at least $m\mathrm{/}2$ and lies in $\mathrm{int}{D}^2$: the interpolation moves each point by at most $m\mathrm{/}4$ from $u_i(t)$. Hence $\gamma$ lands in $F_n(\mathrm{int}{D}^2)$, and its composition with the quotient map $p$ of [L3] is a continuous map $H:I\times I\to C_n(\mathrm{int}{D}^2)$, $H(s,t):=[\gamma (s,t)]$, with $H(0,t)=\alpha (t)$ and $H(1,t)=[z(t)]$; the identities $\gamma (s,0)=u(0)=Q_n$ and $\gamma (s,1)=u(1)$, together with $[u(1)]=\alpha (1)=[Q_n]$, show that $H(s,0)=H(s,1)=[Q_n]$ for every $s$, so that $H$ is a path homotopy relative to $\{0,1\}$.

The lift of the representative is $z$ itself. The path $[z]:=p\circ z$ is a based loop at $[Q_n]$ by step 2.1, and $z$ is a lift of it with $z(0)=Q_n$; by the uniqueness clause [L4] the unique ordered lift of $[z]$ from $Q_n$ is exactly $z$. Together with steps 1.2, 2.1 and 3.1 this exhibits the required smooth collision-free lift with flat time ends and the path homotopy $\alpha \simeq [z]$ rel $\{0,1\}$; all constructions used only the given loop, fixed polynomials and the fixed step function, so no choice principle is spent.