Algebraic open inclusions surject on fundamental groups

Charts and path-connected neighbourhoods. The regular local rings of $S$ are domains by [F7]. The minimal-prime correspondence shows that only one irreducible component passes through any point; the finitely many components are therefore disjoint and both closed and open. Connectedness leaves one component. We may consequently view $S$ as an integral finite-type separated scheme by the classical-scheme equivalence in [F7]; smoothness of the algebraic variety means every local ring is regular, so by the equivalence of [F7] the structure morphism $S\to \mathrm{Spec}\mathbb{C}$ is scheme-smooth. Fix $x\in S$. Applying the Jacobian criterion of [F7] to this morphism gives an affine neighbourhood of $x$ with a presentation $C_h\cong (\mathbb{C}[t_1,\dots ,t_m]\mathrm{/}(f_1,\dots ,f_r){)}_g$ in which some $r\times r$ minor of the Jacobian matrix $(\mathrm{\partial }{f}_j\mathrm{/}\mathrm{\partial }{t}_i)$ is invertible at $x$, and whose fibres have dimension $m-r$; here the fibre is the local variety itself, so its chart dimension is $m-r$, and the nonempty-open dimension equality of [F7] gives $m-r=n=\dim S$. If $r=0$ the chart is already open in affine space. For $r>0$, after relabelling the coordinates and applying the holomorphic implicit function theorem of [F7] (adding an unused parameter if the free-coordinate block is empty) to the holomorphic map $(f_1,\dots ,f_r)$ near the image of $x$, the common zero locus is locally the graph of a holomorphic function of the remaining $m-r=n$ coordinates; the graph coordinate map identifies a neighbourhood of $x$ in the analytic topology with an open ball in ${\mathbb{C}}^n$, and this analytic topology is the one of [F7]. The same presentation and implicit-function argument applied to the reduced induced structure of a locally closed smooth complex subvariety exhibits its points as complex charts of dimension equal to its complex dimension. Consequently $S$, and every smooth locally closed complex subvariety of $S$, is a complex manifold for the analytic topology, hence in particular a smooth real manifold of twice its complex dimension; the chart balls are path-connected neighbourhoods, so $S$ and its open subsets are locally path connected.

Smoothing a based loop. Let $\gamma :S^1\to S$ be a continuous based loop. Choose a continuous surjection $\psi :S^1\to S^1$ with $\psi (\ast )=\ast$ that is constant on a neighbourhood of $\ast$: in angular coordinates it is induced by a nondecreasing surjection $\lambda :[0,1]\to [0,1]$ constant at $0$ and $1$ on endpoint collars. The interpolation $(1-u)t+u\lambda (t)$ descends to a homotopy from the identity fixing $\ast$; then ${\gamma }_1=\gamma \circ \psi$ is a continuous based loop, constant on an open arc $W_0\ni \ast$ where ${\gamma }_1=s$, and it is based-homotopic to $\gamma$. The map ${\gamma }_1$ is smooth (indeed constant) on an open neighbourhood of the closed set $A=\{\ast \}$, so the relative Whitney approximation of [F5] applies: there are a smooth based loop ${\gamma }_2:S^1\to S$ and an open neighbourhood $W$ of $\ast$ such that ${\gamma }_2={\gamma }_1$ on $W$ and ${\gamma }_1$ is homotopic to ${\gamma }_2$ by a homotopy fixed on $W$. Thus ${\gamma }_2$ is a smooth based loop, equal to ${\gamma }_1=s$ on the open neighbourhood $W\cap W_0$ of the base point, with $[{\gamma }_2]=[{\gamma }_1]=[\gamma ]$ in ${\pi }_1(S,s)$.

A finite decomposition of the complement into codimension-two smooth pieces. The complement $Z=S\setminus U$ is closed and proper, and $S$ is irreducible by step 1.1. We construct pairwise disjoint smooth locally closed complex subvarieties $(Z_j)$ covering $Z$, each of complex dimension at most $n-1$, by downward induction on dimension. Every irreducible component $C$ of $Z$ is a proper closed irreducible subvariety of the irreducible $S$, so $\dim C\le n-1$: append $S$ to any chain of irreducible closed subsets of $C$ to obtain a chain one term longer in $S$, using the chain-dimension definition in [F7]. The same argument gives the dimension drop for every proper closed subset of an irreducible component below. Suppose a closed algebraic subset $T\subseteq Z$ is given in which every irreducible component has dimension at most $d$. For each irreducible component $C$ of $T$ of dimension exactly $d$, the regular locus $C_{\mathrm{r}\mathrm{e}\mathrm{g}}$ is a nonempty Zariski open subset of $C$: the generic-smoothness corollary of [F7], applied to the constant morphism from the irreducible classical variety $C$ to a point, produces a nonempty open subset of the regular locus, which is open by [F7]. The sets $C_{\mathrm{r}\mathrm{e}\mathrm{g}}\setminus {\bigcup }_{C^{\mathrm{\prime }}\mathrm{\ne }C}{C}^{\mathrm{\prime }}$, for the finitely many top-dimensional components $C$, are pairwise disjoint smooth locally closed complex subvarieties of dimension $d$. What they do not cover of $T$ is the union of the lower-dimensional components, the singular loci of the top-dimensional components, and the intersections of a top-dimensional component with another component; the lower-dimensional components have dimension at most $d-1$ by assumption and the other sets are proper closed subsets of top-dimensional irreducible components, hence have dimension at most $d-1$ by the chain argument, so the leftover $T^{\mathrm{\prime }}$ is a closed algebraic subset all of whose components have dimension at most $d-1$. Iterating from $T=Z$ and $d=n-1$ produces finitely many disjoint smooth locally closed pieces, and the iteration stops at dimension $0$, where the pieces are points and nothing is left over. Each piece has complex dimension at most $n-1$, so its real dimension is at most $2n-2$ and its real codimension in the smooth real manifold $S$ of step 1.1 is at least two; being a smooth locally closed complex subvariety, each piece is an embedded submanifold of $S$.

A cutoff vanishing at the base point and equal to one over the bad set. Let ${\gamma }_2$ be the smooth loop of step 1.2, denoting by ${\gamma }_2^{-1}(Z)\subseteq S^1$ the compact bad set. If it is empty, then ${\gamma }_2$ already lies in $U$ and its class is represented in $U$; assume it is nonempty. Since ${\gamma }_2$ equals the constant $s\in U$ on the open neighbourhood $W_2=W\cap W_0$ of $\ast$, the compact set $K={\gamma }_2^{-1}(Z)$ is disjoint from $W\cap W_0$. Choose an open arc $W^{\mathrm{\prime }}\ni \ast$ with $\overline{W^{\mathrm{\prime }}}\subseteq W_2$, put $U_1=S^1\setminus \overline{W^{\mathrm{\prime }}}$ and $U_2=W_2$; these open sets cover $S^1$, and $K\subseteq S^1\setminus W_2\subseteq U_1$. By the partition-of-unity theorem [F4] applied to the cover $\{U_1,U_2\}$ choose a subordinate smooth partition of unity $({\chi }_1,{\chi }_2)$ and set $\chi ={\chi }_1$. Then $\chi$ is smooth with values in $[0,1]$, equal to $1$ on the open set $S^1\setminus \mathrm{supp}({\chi }_2)\supseteq S^1\setminus W_2\supseteq K$, and equal to $0$ on the open set $S^1\setminus \mathrm{supp}({\chi }_1)\supseteq S^1\setminus U_1=\overline{W^{\mathrm{\prime }}}\ni \ast$; moreover whenever $\chi (p)=0$ we have ${\chi }_2(p)=1$, so $p\in \mathrm{supp}({\chi }_2)\subseteq W_2$ and hence ${\gamma }_2(p)=s\in U$. In particular $\chi (\ast )=0$.

The truncated perturbation family is transverse to every piece. Apply the perturbation lemma [F3] to the smooth loop ${\gamma }_2:S^1\to S$: there are an open ball $B\subseteq {\mathbb{R}}^m$, positive-dimensional as recorded in [F3] because $S\mathrm{\ne }\mathrm{\varnothing }$, and a smooth family $\mathcal{F}:S^1\times B\to S$ with ${\mathcal{F}}_0={\gamma }_2$ and with every parameter map $a\mapsto \mathcal{F}(p,a)$ a submersion; in particular the evaluation map $\mathcal{F}$ is a submersion, hence transverse to every embedded submanifold by [F2]. Define $\tilde{\mathcal{F}}:S^1\times B\to S$ by $\tilde{\mathcal{F}}(p,a)=\mathcal{F}(p,\chi (p)a)$; since the ball $B$ is centred at $0$ and $0\le \chi (p)\le 1$ this is well defined and smooth, and ${\tilde{\mathcal{F}}}_0={\mathcal{F}}_0={\gamma }_2$. For every $a\in B$ one has $\tilde{\mathcal{F}}(\ast ,a)=\mathcal{F}(\ast ,\chi (\ast )a)=\mathcal{F}(\ast ,0)={\gamma }_2(\ast )=s$, so every slice ${\tilde{\mathcal{F}}}_a$ is a based loop. At a point $(p,a)$ with $\chi (p)>0$ the derivative of $\tilde{\mathcal{F}}$ in the parameter direction is $\chi (p)$ times the parameter derivative of $\mathcal{F}$ at $(p,\chi (p)a)$, which is surjective, so $\tilde{\mathcal{F}}$ is a submersion there. Hence $\tilde{\mathcal{F}}⋔Z_j$ for every piece $Z_j$ of step 2.1: if $\tilde{\mathcal{F}}(p,a)\in Z_j\subseteq Z$ then $\chi (p)>0$, since $\chi (p)=0$ would give $\tilde{\mathcal{F}}(p,a)=\mathcal{F}(p,0)={\gamma }_2(p)$ with ${\gamma }_2(p)\in U$ by step 2.2, and at a point with $\chi (p)>0$ the map is a submersion and therefore transverse by [F2].

A parameter outside all bad sets, and surjectivity. For each piece $Z_j$ of step 2.1 the set $N_j\subseteq B$ of parameters $a$ with ${\tilde{\mathcal{F}}}_a$ not transverse to $Z_j$ is null by the parametric transversality theorem [F2]. Since there are finitely many pieces, the union $N={\bigcup }_j{N}_j$ is null in the positive-dimensional smooth manifold $B$ by [F6], so its complement is dense by [F6] and in particular nonempty; choose $a\in B\setminus N$. Then the slice ${\tilde{\mathcal{F}}}_a:S^1\to S$ is a smooth based loop transverse to every $Z_j$. A transverse map from a one-dimensional manifold cannot meet a piece of real codimension at least two: if ${\tilde{\mathcal{F}}}_a(p)\in Z_j$, transversality would give $\dim T_{{\tilde{\mathcal{F}}}_a(p)}S\le \dim \mathrm{im}(d({\tilde{\mathcal{F}}}_a{)}_p)+\dim T_{{\tilde{\mathcal{F}}}_a(p)}{Z}_j\le 1+(2n-2)<2n$, a contradiction. Hence ${\tilde{\mathcal{F}}}_a(S^1)$ meets no $Z_j$ and therefore lies in $U$. The map $(p,u)\mapsto \tilde{\mathcal{F}}(p,ua)$, $0\le u\le 1$, is a homotopy from ${\gamma }_2$ to ${\tilde{\mathcal{F}}}_a$ through based loops, so $[{\tilde{\mathcal{F}}}_a]=[{\gamma }_2]=[\gamma ]$ in ${\pi }_1(S,s)$ by [F8], while $[{\tilde{\mathcal{F}}}_a]$ lies in the image of ${\pi }_1(U,s)\to {\pi }_1(S,s)$ by [F8]. In the discarded case $K=\mathrm{\varnothing }$ of step 2.2 the loop ${\gamma }_2$ itself already lies in $U$, so the same conclusion holds. Since every class in ${\pi }_1(S,s)$ has a continuous based loop representative, the map ${\pi }_1(U,s)\to {\pi }_1(S,s)$ is surjective.

Path connectedness. Let $x,y\in U$. The chart balls of step 1.1 make $S$ locally path connected, so connectedness and [F9] give a continuous path $\beta$ from $x$ to $y$; reparametrize it to be constant near its endpoints. Traverse $\beta$ on the first quarter of $S^1$, its reverse on the second quarter, and the constant path at $x$ on the second half. This gives a continuous circle map constant near all junctions. Apply [F5] to this circle map, relative to the finite closed set of junctions, to obtain a smooth loop equal to the original on neighbourhoods of those junctions. Its bad set is disjoint from those neighbourhoods. Repeat step 2.2 with $W_2$ the union of these constant neighbourhoods and $W^{\mathrm{\prime }}$ a smaller union with closure in $W_2$. The same two-set partition supplies a cutoff vanishing near all junctions, equal to one near the bad set, and with every zero lying where the loop is in $U$. Steps 3.1 and 4.1 then give a perturbed loop in $U$ agreeing with the original near the junctions. Its first quarter joins $x$ to $y$ in $U$. Thus $U$ is path connected; when $Z=\mathrm{\varnothing }$ the unperturbed path already suffices.