Isotopy invariance and nonrepeating link-homotopy scope of Milnor invariants

Let ${\tau }_t$ be the given smooth isotopy through string links, fixed on common endpoint collars. Reparametrize it to be stationary near $t=0,1$. Choose fixed disjoint smooth closure arcs outside $C=D^2\times [0,1]$, using the closure construction in the based-string-link definition, and adjoin them to the moving intervals. The result is a smooth family of closed links in $S^3$, constant on the closure arcs and endpoint collars; extend it constantly in time outside $[0,1]$. The map $\mathrm{\Theta }:\mathbb{R}\times {⨆}_i{S}^1\to \mathbb{R}\times S^3$, $\mathrm{\Theta }(t,s)=(t,{\bar{\tau }}_t(s))$, is an injective immersion and is proper: injectivity and immersion follow from the time coordinate and the embedding of each link slice, while the preimage of a compact set lies in a compact time interval times the finite union of circles. Thus its image is a closed embedded submanifold. The vector field $Y(t,{\bar{\tau }}_t(s))=(0,{\mathrm{\partial }}_t{\bar{\tau }}_t(s))$ along this graph is smooth and vanishes on the fixed closure arcs and endpoint collars. Since the motion is confined to a compact subinterval of each string component away from its endpoint collars, choose a compact set $K\subset \mathrm{int}C$ containing every point where this velocity can be nonzero. By [A1] and [L1], [F4] extends $Y$ to a global smooth vector field on $\mathbb{R}\times S^3$. Let $X_t$ be its $S^3$ component. By [F5] choose $\chi :S^3\to [0,1]$ equal to one near $K$ and supported in $\mathrm{int}C$, and replace $X_t$ by $\chi X_t$. This field still equals the prescribed velocity along the entire closed link: it is unchanged wherever the velocity is nonzero and is zero on the stationary arcs. Its supports lie in the compact set $\mathrm{supp}\chi \subset \mathrm{int}C$. Integrate it by [F6]. Uniqueness of the flow gives an ambient isotopy $H_t$ with $H_t{\bar{\tau }}_0={\bar{\tau }}_t$; its support in $\mathrm{int}C$ makes $H_t$ the identity near $\mathrm{\partial }C$, so it preserves $C$ and fixes the endpoint collars, bottom basepoint, and side connector. This is the relative based isotopy-extension argument for string links.

Under [A1], an ambient isotopy of closed links induces the peripheral correspondence in [F3]. Applying that correspondence to the based word-lift data gives the transported based residue candidate. The first assertion of [F2] proves independence of every meridian basing and finite longitude-word lift, with exactly equal lower-coefficient ideals. Applying it to the transported word data proves that the two closed-link residues agree in the same residue group.

In the string-link case, take the target peripheral datum to be the transported datum required in the statement. Restricting $H_1$ gives a based homeomorphism of string-link exteriors that transports the selected tubes, positive meridians, parallels, and access paths. It fixes the endpoint basepoints and side connector, so the two end inclusions commute with the induced based group isomorphisms. By [F1], the longitude has a unique preimage in each finite bottom-end quotient; naturality carries that preimage to the corresponding class for the isotopic string link. The Magnus expansion is the same homomorphism on the transported meridian symbols, so every integer coefficient agrees.

Under [A1], a link-homotopy is generated by ambient isotopies and self-crossing changes. For pairwise-distinct indices, ambient isotopy is handled by step 1.2 and each self-crossing change is the second assertion of [F2]. Each move identifies the lower-coefficient ideals by [F2]; composing these finitely many equalities proves the asserted residue equality. No repeated-index conclusion is drawn.