Finite-parameter smoothing of vector-bundle sections

If $X$ is empty, take the unique empty family. If $M$ is empty, its bundle has the unique empty smooth section, and take the unique family of those sections. In either case all agreement constraints and the straight-line homotopy hold vacuously, and the displayed error is $\sup \{0\}=0$. We may therefore assume $X$ and $M$ nonempty. Choose a finite bundle atlas and a subordinate finite smooth partition of unity using compactness and [F1, F2]. In each chart multiply a section coefficient by its partition function, extend by zero within the chart, convolve with a fixed compactly supported smooth mollifier of radius $\delta$, and sum the finitely many resulting local sections. For each $\delta >0$ this defines a linear operator $S_{\delta }:C^0(M,E)\to {\mathrm{\Gamma }}^{\mathrm{\infty }}(E)$: every output derivative is an integral of the input against a derivative of a fixed smooth kernel, so each $C^r$ seminorm is bounded by a finite constant times the input $C^0$ norm. Uniform continuity of each input on compact chart supports gives $\mathrm{\parallel }{S}_{\delta }u-u{\mathrm{\parallel }}_{C^0}\to 0$ for every $u$. Since $x\mapsto s(x,\cdot )$ has compact image in $C^0(M,E)$, a finite subcover argument makes this convergence uniform over $x\in X$.

The family $x\mapsto S_{\delta }s(x,\cdot )$ is continuous into $C^{\mathrm{\infty }}$ by the seminorm bounds in step 1.1. Let $U$ be the chosen collar neighborhood of $K$, with projection $r:U\to K$, and choose a continuous cutoff $\chi :X\to [0,1]$ equal to $1$ on $K$ and supported inside $U$. For an interior hypersurface use the two-sided collar $c:K\times (-2a,2a)\to U$, $r(c(k,t))=k$, with $\chi$ supported in $\mathrm{\mid }t\mathrm{\mid }<a$; no boundary of $X$ is needed. Set $s_1(x)=S_{\delta }s(x)+\chi (x)(s(r(x))-S_{\delta }s(r(x)))$ on $U$ and $s_1=S_{\delta }s$ off $U$. Every term is a smooth section in $m$ and varies continuously in $C^{\mathrm{\infty }}$ with $x$. At $x\in K$, $s_1(x)=s(x)$. As $x$ approaches $K$, continuity into $C^0$ and the uniform approximation in step 1.1 make the correction uniformly small; choose the collar width and then $\delta$ so $s_1$ remains arbitrarily $C^0$-close to $s$.

For each marked point choose a bundle chart and a smooth bump $b_i$ supported in that chart with $b_i(m_i)=1$; shrink the charts to be pairwise disjoint. In the chart extend $v_i-s_1(x,m_i)$ as the constant coefficient vector and multiply by $b_i$. The resulting correction $c_i(x,\cdot )$ is a smooth section continuous in $x$ in every $C^r$ seminorm, is zero on $K$ because $s_1=s=v_i$ there, and changes the value at $m_i$ by exactly $v_i-s_1(x,m_i)$. Put $\tilde{s}=s_1+{\sum }_i{c}_i$. The finite corrections preserve $C^{\mathrm{\infty }}$ continuity and the relative condition. Since the errors at the marked points are bounded by the uniform $C^0$ error from step 2.1 and the bumps have fixed bounded norms, choosing that error sufficiently small gives the prescribed $\epsilon$ bound.

For $0\le t\le 1$ define $H_t(x,m)=(1-t)s(x,m)+t\tilde{s}(x,m)$ in the vector space $E_m$. This is continuous, equals $s$ at $t=0$ and $\tilde{s}$ at $t=1$, and is constant on $K$ and at each marked point because both endpoint families agree there. Thus the required relative homotopy exists.