Existence and isotopy invariance of the double branched cover of a link

Compose the Hurewicz isomorphism in [F2], the Alexander-duality isomorphism in [F1], and coordinate-sum modulo two on ${\mathbb{Z}}^r$. The local intersection description in [F1] sends a meridian of $K_i$ to $\pm e_i$, whose coordinate sum is $1$ modulo two. This defines ${ϵ}_L$ and proves it is surjective because $r\ge 1$; hence its kernel has index two.

An ambient isotopy from $L$ to $L^{\mathrm{\prime }}$ restricts to an orientation-preserving diffeomorphism of their complements and carries each meridian class to a meridian class. It therefore carries $\ker {ϵ}_L$ to $\ker {ϵ}_{L^{\mathrm{\prime }}}$.

By [F3] and [F4], the index-two kernel determines a connected two-sheeted cover. On a punctured tube around $K_i$, its two monodromies are the meridian value $1$ and a longitude value ${\eta }_i\in {\mathbb{F}}_2$. The local model [F5] has exactly these monodromies, so covering classification identifies its punctured part with the restricted cover. Attach its core to obtain a smooth extension in the chosen charts. This constructs an extension; independence of the smooth total-space diffeomorphism type requires the normalization below, not merely equality of boundary monodromies.

Compare two oriented tubular charts ${\phi }_0,{\phi }_1$ around the same component. Express them over a common oriented core parameter $s$; a change of the original core parameter only replaces the unit phase $e^{\mathrm{i}{\eta }_{j}s}$ by a smooth unit loop $a_j(s)$ of the same winding number ${\eta }_j$. Let $A(s)\in G{L}^{+}(2,\mathbb{R})$ be the unique vertical linear change for which ${\phi }_1\circ A$ and ${\phi }_0$ induce the same normal-quotient identification. Write $A=RP$ by smooth polar decomposition, with $R(s)\in SO(2)$ and $P(s)$ positive definite. The compatible-tube assertion [F6] gives an ambient isotopy fixing the core and carrying ${\phi }_0$ to ${\phi }_1\circ A$ near it. The fiber-linear isotopy $R((1-t)P+tI)$ then connects $A$ to $R$ through invertible maps. On a common smaller tube its composition with ${\phi }_1$ is an isotopy of open embeddings, with open track, fixing the core. The relative open-set clause of [F6] extends it near the compact core. Thus an ambient diffeomorphism $g$, isotopic to the identity through maps fixing the whole link, satisfies $g\circ {\phi }_0={\phi }_1\circ R$ near this component. Choose the supports in disjoint component tubes and compose the finitely many isotopies.

Identify $R(s)$ with a unit complex number, of winding $k$. The image under $g$ of the first chart longitude has slope ${\lambda }_1+k\mu$ in the second chart, so preservation of the meridian character gives ${\eta }_0\equiv {\eta }_1+k(\mathrm{m}\mathrm{o}\mathrm{d}2)$. The unit loop $R(s)a_0(s)\mathrm{/}{a}_1(s)$ therefore has even winding. It has a smooth periodic square root $r(s)$: lift its argument on $[0,2\pi ]$, halve it, and use even winding to match the endpoints and derivatives. In the two branch models $b_j(s,z)=(s,a_j(s)z^2)$ the map $(s,z)\mapsto (s,r(s)z)$ consequently covers $g$, since $a_1{r}^2=R{a}_0$. It is an orientation-preserving smooth diffeomorphism including $z=0$. The two signs of $r$ differ by the deck map. Choose the sign matching the complement-cover isomorphism at one point of each connected punctured tube; covering uniqueness makes them agree throughout that tube. Hence the local lifts glue to the complement lift. This proves independence of the oriented smooth total-space diffeomorphism type from arbitrary oriented tubular charts. It does not assert a smooth lift over the identity base map between arbitrary unnormalized charts.

Let $F$ be the terminal diffeomorphism of the given ambient isotopy from $L$ to $L^{\mathrm{\prime }}$. First choose the target tubular charts by transporting the source charts through $F$. Corresponding longitudes have the same character, and in these transported branch coordinates a lift is $(s,z)\mapsto (s,z)$, or its deck translate. By step 1.2 and [F4] the complement covers are isomorphic over $F$; on every punctured tube that isomorphism is one of these two local lifts, so it extends smoothly across the branch core. The resulting map is an orientation-preserving diffeomorphism of the closed total spaces. Finally steps 3.1–4.1 compare the transported target charts with any originally chosen target charts. Thus $\mathrm{\Sigma }(L^{\mathrm{\prime }})\cong \mathrm{\Sigma }(L)$ as oriented manifolds, with no unjustified smooth extension of an arbitrary normal germ.