Relative finite-parameter avoidance of reducible connections

If $K=\mathrm{\varnothing }$ or $f(K)\subseteq {\mathcal{A}}_{\mathrm{\infty }}^{\ast }$, the constant homotopy suffices. Otherwise realize the given finite simplicial complex by its barycentric coordinates as a compact subset of ${\mathbb{R}}^q$, taking $q\ge 1$. Let $U=f^{-1}({\mathcal{A}}_{\mathrm{\infty }}^{\ast })$, an open neighborhood of $L$ by [F2]. Since $K$ is compact metrizable and $L$ is a closed subcomplex, choose nested relatively open metric neighborhoods $U_0,U_1,U_2$ of $L$ with ${\bar{U}}_0\subset U_1$, ${\bar{U}}_1\subset U_2$, and ${\bar{U}}_2\subset U$; if $L=\mathrm{\varnothing }$ take them empty. The set $f({\bar{U}}_2)$ is compact because it is the continuous image of a compact set. The inclusion from the $C^{\mathrm{\infty }}$ section topology to the finite-atlas $C^1$ topology is continuous, so this image is also compact in the $C^1$ topology. For each of its members, [F2] supplies a $C^1$ neighborhood contained in ${\mathcal{A}}_{\mathrm{\infty }}^{\ast }$. Cover the compact image by the half-radius neighborhoods; a finite subcover and the minimum of their positive half-radii give one $C^1$ tolerance such that every pointwise perturbation smaller than that tolerance remains irreducible. Include the corresponding $C^1$ seminorms among the finitely many seminorms in which all forthcoming approximations are to be small.

Fix a small positive tolerance $\delta$ in each of those seminorms. Uniform continuity on compact $K$ gives a radius $\rho >0$ so that $f(s_i)-f(s)$ is smaller than $\delta$ in every selected seminorm when $s,s_i\in K$ and $\mathrm{\mid }s-s_i\mathrm{\mid }<2\rho$. Choose finitely many $s_i\in K$ with the balls $B_{\rho }(s_i)$ covering $K$. Let ${\psi }_i$ be nonnegative smooth ambient bumps supported in $B_{2\rho }(s_i)$ and positive on $B_{\rho }(s_i)$, constructed by translating and rescaling $\exp (-1\mathrm{/}(1-\mathrm{\mid }x{\mathrm{\mid }}^2))$ on the unit ball and extending by zero. On the open neighborhood $V=\{{\sum }_i{\psi }_i>0\}$ of $K$, put ${\lambda }_i={\psi }_i\mathrm{/}{\sum }_j{\psi }_j$ and $g(s)=A_0+{\sum }_i{\lambda }_i(s)(f(s_i)-A_0)$. This is an ambient smooth finite-affine family of smooth connections. For $s\in K$, every contributing sample is within $2\rho$ of $s$, so $p(g(s)-f(s))<\delta$ for each selected seminorm $p$. This construction smooths the parameter dependence directly and simultaneously on the whole polyhedron; it changes no face independently of its cofaces.

Choose a smooth ambient cutoff $0\le \theta \le 1$ which is zero on a neighborhood of ${\bar{U}}_0$ and one on a neighborhood of the disjoint compact set $K\setminus U_1$. Such a cutoff follows from finitely many ambient bumps separating the two compact sets. Define on $K$ the continuous family $h(s)=f(s)+\theta (s)(g(s)-f(s))$. The straight interpolation from $f$ to $h$ is fixed near $L$ and is $\delta$-small in every selected seminorm. By the uniform $C^1$ margin in step 1.1, choosing $\delta$ small ensures it is irreducible on ${\bar{U}}_2$. Its closed reducible-parameter set $R=\{s\in K:h(s)\mathrm{\notin }{\mathcal{A}}_{\mathrm{\infty }}^{\ast }\}$ is compact and contained in $K\setminus U_2$, hence in the ambient open region where $\theta =1$ and $h=g$ on $K$. If $R$ is empty the first interpolation already proves the result. Otherwise choose an ambient open $W$ with compact closure in $V\cap \{\theta =1\}$, containing $R$ and avoiding a neighborhood of $L$, and a smooth cutoff $0\le \chi \le 1$ with compact support $T\subset W$, positive on $R$. All subsequent holonomy calculations use the ambient family $g$ on $W$; on the actual parameter subset $K\cap W$ this is exactly $h$.

Choose $r\ge 2$ so that $2(r-1)(N-1)>q$, where $q$ is the ambient parameter dimension in step 1.1. Put $d=\dim M\in \{3,4\}$ and use stereographic projection from $m$ to identify $M\setminus \{m\}$ smoothly with ${\mathbb{R}}^d$. Choose $r$ pairwise disjoint closed Euclidean coordinate balls there. Their complement is path connected: choose pairwise disjoint slightly larger balls around them, and replace every portion of a polygonal path that enters one of the larger balls by a path on its boundary sphere. These replacement spheres lie outside the original closed balls, and they are connected because $d\ge 2$. A ray outside a large ball then extends to $m$ under inverse stereographic projection, so exterior paths can reach the basepoint. In each ball choose a small embedded circle and a short closed arc on it. Join $m$ to a point on the circle outside that arc by a piecewise-smooth stem disjoint from that arc and meeting no other ball; return along the same stem after traversing the circle once. This gives based piecewise-smooth loops ${\mathrm{\ell }}_i$. In the Euclidean ball coordinates, a short circle arc has an explicit product tube: after a rigid coordinate change write the circle as $\gamma (t)=a(\cos t,\sin t,0,\dots ,0)$ for $\mathrm{\mid }t\mathrm{\mid }<\alpha <\pi$, and use $(t,z)\mapsto (a+z_1)(\cos t,\sin t,0,\dots ,0)+z_2{e}_3+\cdots +z_{d-1}{e}_d$ for sufficiently small $z$. Choose the arc away from its stem endpoint and shrink this tube so it meets the loop only on the arc and misses every stem. The disjoint balls make the tubes pairwise disjoint, so each loop has a private control tube.

Parametrize each loop as ${\mathrm{\ell }}_i:[0,1]\to M$ and subdivide it into finitely many smooth segments, each lying in a smooth bundle trivialization. Let $U_i(s,t)$ denote transport from the chosen base frame to the point ${\mathrm{\ell }}_i(t)$. On a segment its matrix equation is $U^{\mathrm{\prime }}=-B_i(s,t)U$; by the realification in [F4], left multiplication by $-B_i(s,t)$ is a real $2N^2\times 2N^2$ coefficient matrix smooth in the parameter $s\in W$ and segment variable $t$. For each fixed $s_0\in W$, the local solution intervals supplied by [F4] cover the compact segment. A finite subcover, a subdivision subordinate to it, and the intersection of its finitely many parameter neighborhoods give smooth solution maps across the whole segment near $s_0$. Since this is local in $s_0$, it gives smooth parameter dependence throughout $W$; finite composition with the smooth bundle transition matrices preserves smoothness across segment endpoints. In tube coordinates $(t,z)$ around the private arc, choose the loop parametrization to agree with $t$ along this arc, and choose a bump ${\rho }_i(t)$ of integral one supported away from the arc ends, a transverse bump $c(z)$ with $c(0)=1$, and a real basis $X_a$ of $\mathfrak{s}\mathfrak{u}(N)$. In a fixed smooth frame on the tube define the adjoint-valued form $b_{i,a}(s)={\rho }_i(t)c(z)U_i(s,t)X_a{U}_i(s,t{)}^{-1}dt$, extending its coefficients constantly in the transverse coordinates before multiplying by $c$ and extending by zero outside the tube. It is a genuine smooth bundle-valued one-form depending smoothly on the ambient parameter. Along the loop its contraction is exactly the displayed transported Lie-algebra element.

The realified equation in [F4] applies to the $w$-dependent coefficient $B_i(s,t)+{\sum }_{j,a}{w}_{j,a}{b}_{j,a}(s,t)$ on each loop segment, so $\mathrm{\Psi }$ is a smooth real map and its parameter derivative exists. For the full transport $U:[0,1]\to SU(N)$ satisfying $U^{\mathrm{\prime }}=-B(t)U$ on each segment and a perturbation $b(t)$, differentiate the real matrix ODE segmentwise to obtain $(U^{-1}\delta U{)}^{\mathrm{\prime }}=-U^{-1}bU$ and integrate across the finitely many segments to get $U(1{)}^{-1}\delta U(1)=-{\int }_0^{1}U(t{)}^{-1}b(t)U(t)dt$. The control form in step 5.1 makes this last integral $-X_a$ for loop $i$. The other loop holonomies do not vary, since their images miss its tube. Therefore, for $D=r(N^2-1)$ real parameters $w=(w_{i,a})$, the derivative at $w=0$ of $\mathrm{\Psi }(s,w)=({\mathrm{Hol}}_{{\mathrm{\ell }}_i}(g(s)+{\sum }_{j,a}{w}_{j,a}{b}_{j,a}(s)){)}_{i=1}^r$ is an isomorphism in the $w$ variable onto $TSU(N{)}^r$, for every $s\in W$. Smooth ODE dependence and compactness of $T$ give a number $a>0$ on which this derivative remains an isomorphism uniformly for $s\in T$ and $\mathrm{\mid }w\mathrm{\mid }<a$.

Choose a ball $B\subset {\mathbb{R}}^D$ about zero of radius less than $a$ and put $\mathrm{\Phi }(s,v)=\mathrm{\Psi }(s,\chi (s)v)$ for $s\in W$. Where $\chi (s)>0$, its parameter derivative is $\chi (s)D_w\mathrm{\Psi }(s,\chi (s)v)$, still invertible by step 6.1. For each $1\le k<N$, the map ${\mathcal{F}}_k:((W_{+}\times {\mathrm{Gr}}_{\mathbb{C}}(k,N))\times B)\to SU(N{)}^r\times {\mathrm{Gr}}_{\mathbb{C}}(k,N)$, $((s,Q),v)\mapsto (\mathrm{\Phi }(s,v),Q)$, with $W_{+}=\{s\in W:\chi (s)>0\}$, is a submersion and thus transverse to $I_{k,r}$ of [F3]. Notice that the smooth source is the ambient open set $W_{+}$, not an individual simplex or a nonsmooth polyhedron.

By [F5], outside a null subset of $B$ the fixed-$v$ map from $W_{+}\times {\mathrm{Gr}}_{\mathbb{C}}(k,N)$ is transverse to $I_{k,r}$. Its source dimension is $q+2k(N-k)$, while that incidence manifold has codimension $2rk(N-k)$ by [F3]. Since $k(N-k)\ge N-1$ and $2(r-1)(N-1)>q$, the codimension is strictly larger than the source dimension. A transverse slice can therefore have no incidence preimage: a putative preimage would require a surjection from a smaller-dimensional tangent space to the normal space.

If $Z_k=\mathrm{\varnothing }$, its projection to $B$ is empty. Otherwise, in a target chart around each point of $I_{k,r}$, that embedded submanifold is a coordinate slice of codimension $2rk(N-k)$. Since ${\mathcal{F}}_k$ is a submersion, the composite of ${\mathcal{F}}_k$ with the normal-coordinate projection is a submersion; its zero level is the local preimage of $I_{k,r}$. Applying [F7] in these charts shows that $Z_k={\mathcal{F}}_k^{-1}(I_{k,r})$ is an embedded manifold of dimension $q+D+2k(N-k)-2rk(N-k)=q+D-2(r-1)k(N-k)<D$. Every point of its projection to $B$ is critical because $\dim Z_k<D$, so [F8] makes that image null. There are finitely many $k$, and [F9] makes the union of these images and the exceptional sets in step 8.1 null with dense complement. Thus arbitrarily small $v\in B$ avoid every incidence condition throughout $W_{+}$. This also directly verifies the transversality dimension argument by Sard.

Define on all of $K$ the final family $h_v(s)=h(s)+\chi (s){\sum }_{j,a}{v}_{j,a}{b}_{j,a}(s)$, with the added term zero off $W$. The compact support $T$ and its spatial smoothness make this term continuous in every section seminorm, and its value uniformly small in any specified finite list by taking $v$ sufficiently small. If $\chi (s)>0$, then $s\in W$ and $h(s)=g(s)$, so $h_v(s)$ has the holonomy tuple $\mathrm{\Phi }(s,v)$, which has no common invariant proper subspace by step 9.1. Therefore $h_v(s)$ is irreducible. If $\chi (s)=0$, then $s\mathrm{\notin }R$ since $\chi$ is positive on $R$, so $h_v(s)=h(s)$ was already irreducible. The straight perturbation homotopy from $h$ to $h_v$ is fixed on an open neighborhood of $L$. Concatenate it with the interpolation in step 3.1. Choose $\delta$ and then $v$ within half of each requested seminorm tolerance; the triangle inequality gives the asserted bounds relative to the original $f$ for the entire concatenated homotopy. All formulas are global on $K$, so faces and cofaces agree automatically.