Uniform Laurent approximation through bundle automorphisms

Fix a bundle chart over an open $U\subseteq X$ whose closure is compact and lies in a larger chart. For an integer $N\ge 1$, use the Fejér kernel $K_N(t)=N^{-1}\mathrm{\mid }1+e^{it}+\cdots +e^{i(N-1)t}{\mathrm{\mid }}^2$. It is nonnegative, has integral $2\pi$, and expands as ${\sum }_{\mathrm{\mid }j\mathrm{\mid }<N}(1-\mathrm{\mid }j\mathrm{\mid }\mathrm{/}N)e^{ijt}$. Entrywise integration in [F3] therefore defines on $U$ the Laurent polynomial $p_N(x,z)={\sum }_{\mathrm{\mid }j\mathrm{\mid }<N}(1-\mathrm{\mid }j\mathrm{\mid }\mathrm{/}N)c_j(x)z^j$, where $c_j(x)=(2\pi {)}^{-1}{\int }_{-\pi }^{\pi }f(x,e^{it})e^{-ijt}dt$. Riemann-sum convergence uniform on compact chart closures makes every $c_j$ continuous in $x$.

Let $M$ bound the matrix entries of $f$ on the compact chart closure times $S^1$. Given $ϵ>0$, [F2] supplies $\delta >0$ such that $\parallel f(x,z{e}^{-it})-f(x,z)\parallel <ϵ\mathrm{/}2$ for $\mathrm{\mid }t\mathrm{\mid }<\delta$. On $\mathrm{\mid }t\mathrm{\mid }\ge \delta$, $K_N(t)\le (N{\sin }^2(\delta \mathrm{/}2){)}^{-1}$, so the integral of the tail times the bound $2M$ is below $ϵ\mathrm{/}2$ for all sufficiently large $N$. Since $p_N-f$ is the convolution of $f(x,z{e}^{-it})-f(x,z)$ with $K_N\mathrm{/}(2\pi )$, the short-arc and tail estimates prove $p_N\to f$ uniformly on that chart closure.

Choose finitely many such charts and a finite subordinate partition $\{{\varphi }_i\}$ by [F4]. In chart $i$, choose a Laurent approximant $p_i$ within a common tolerance. The section ${\varphi }_i{p}_i$ of $\mathrm{End}(E)$ has support inside its chart and extends by zero; hence $p={\sum }_i{\varphi }_i{p}_i$ is a global finite Laurent polynomial in $z$. Because ${\sum }_i{\varphi }_i=1$, the same tolerance bounds $p-f$ globally.

The automorphisms form an open subbundle of $\mathrm{End}(E)$: in a chart, invertibility is the open condition $\det \mathrm{\ne }0$. Compactness of $X\times S^1$ and the finite chart cover give a positive tolerance such that every section within that tolerance of $f$ is invertible, and every convex combination with $f$ remains within it. Choose $p$ accordingly. Since $f(x,1)=I$, $p(x,1)$ is invertible; put $q(x,z)=p(x,z)p(x,1{)}^{-1}$. Then $q$ is still Laurent polynomial, is normalized, and can be made arbitrarily close to $f$.

The straight-line family $h_s=(1-s)f+sq$ consists of automorphisms by step 4.1, depends continuously on $(x,z,s)$, and satisfies $h_s(x,1)=I$ for every $s$. It is the required normalized homotopy. For the rank-zero bundle the unique family is already polynomial, and for $X=\mathrm{\varnothing }$ every assertion is vacuous.

Let $f_t$ be a normalized automorphism homotopy. Apply steps 1.1–5.1 over the compact parameter space $X\times I$ to obtain a normalized Laurent family $p_t$ uniformly close to $f_t$. If prescribed normalized Laurent approximations $q_0,q_1$ were chosen sufficiently close at the endpoints, the straight segments from $q_0$ to $p_0$ and from $p_1$ to $q_1$ stay in the same open automorphism neighborhood and remain Laurent and normalized. Concatenating these with $p_t$ gives the required normalized Laurent-polynomial homotopy.