The geometric endpoint permutation matches covering monodromy

Given: $n$, the explicit tuple $Q$, a geometric braid class $[\beta ]$ based at $Q$, its point motions and endpoint permutation, the slice loop, and the fixed-basepoint isomorphism $\mathrm{\Phi }$.

The geometric braid classes based at $Q$ form $G_n$, and the endpoint map ${\pi }_{\mathrm{g}\mathrm{e}\mathrm{o}}:G_n\to S_n$ is a well-defined homomorphism; on a braid representative, $z_j(1)=q_{{\pi }_{\mathrm{g}\mathrm{e}\mathrm{o}}(\beta )(j)}$ (The isotopy classes of geometric braids based at $Q$ form a group, and the endpoint permutation is a homomorphism, Geometric braids in the disc with setwise endpoints).

The slice $S(\beta )(t)=[(z_1(t),\dots ,z_n(t))]$ is a continuous based loop in $C_n(\mathrm{int}{D}^2)$ at $[Q]$ (A geometric braid slices to an interior configuration loop).

A based loop at $[Q]$ has a unique lift to $F_n(D^2)$ starting at $Q$ (Endpoint monodromy of an unordered configuration loop as a permutation of the labels).

The open-to-closed inclusions induce maps ${\iota }_{\ast }^F$ and ${\iota }_{\ast }^C$ at the same basepoints, and the quotient square commutes: ${\iota }^C\circ p_{\mathrm{int}{D}^2}=p_{D^2}\circ {\iota }^F$ (The interior-disc and closed-disc configuration spaces are homotopy equivalent).

The induced map on fundamental groups satisfies $f_{\ast }([\alpha ])=[f\circ \alpha ]$ (The homomorphism on fundamental groups induced by a pointed continuous map).

The fundamental group is a group under the first-loop-then-second product, with identity and two-sided inverses (Based loops and the fundamental group, Loop classes form the group ${\pi }_1(X,x_0)$ under concatenation).

Each coordinate motion is continuous, remains in the open disc, and is pairwise distinct from the others. The coordinate tuple therefore defines a continuous path into $F_n(\mathrm{int}{D}^2)$; continuity into the product follows because the product topology is generated by projection preimages, and distinctness puts the tuple in the ordered-configuration subspace (Geometric braids in the disc with setwise endpoints, Based motions of an unordered point configuration, The product set ${\prod }_{i\in I}{X}_i$ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space, Ordered configuration spaces $F_n(X)$).

For $n=0$ there is one empty geometric braid and $S_0$ is trivial (Geometric braids in the disc with setwise endpoints, Endpoint monodromy of an unordered configuration loop as a permutation of the labels).

At the fixed basepoint $Q$, $\mathrm{\Phi }([\beta ])=({\iota }_{\ast }^C[S(\beta )]{)}^{-1}$ is the isomorphism from $G_n$ to $B_n^{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{f}}$ (Geometric braid classes and the unordered configuration fundamental group).

The endpoint monodromy ${\pi }_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{f}}:B_n^{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{f}}\to S_n$ is a group homomorphism (Endpoint monodromy of an unordered configuration loop as a permutation of the labels).

If the terminal tuple of a lifted loop is recorded by $\tilde{\alpha }(1{)}_i=q_{e(i)}$, then its endpoint record satisfies $e={\sigma }^{-1}$ (Endpoint monodromy of an unordered configuration loop as a permutation of the labels).

For $n=1$ every geometric braid has identity endpoint permutation, and $S_1$ is trivial (Geometric braids in the disc with setwise endpoints, Endpoint monodromy of an unordered configuration loop as a permutation of the labels).

If the unique lift of a based loop at $[Q]$ from $Q$ ends at $\sigma \cdot Q$, then its endpoint monodromy is ${\pi }_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{f}}([\alpha ])=\sigma$ (Endpoint monodromy of an unordered configuration loop as a permutation of the labels).

The tuple $Q$ and every coordinate path are specified. The ordered lift used below is the unique lift from $Q$, so the Axiom of Choice is not used.