A compact smooth manifold has a finite good-cover nerve model

The neighborhoods in [F1] cover $F$; compactness gives a finite subcover $U_1,\dots ,U_N$. Each nonempty finite intersection remains strongly geodesically convex by [F1]. The nerve $K$ has finitely many vertices and faces, so it is a finite simplicial complex. If $F=\mathrm{\varnothing }$, the empty cover and empty realization give the stated boundary case.

Choose a smooth partition of unity ${\varphi }_1,\dots ,{\varphi }_N$ subordinate to the finite cover by [F2]. Define $f:F\to \mathrm{\mid }K\mathrm{\mid }$ by $f(x)={\sum }_i{\varphi }_i(x)v_i$, where $v_i$ is the $i$th vertex. The active indices $\{i:{\varphi }_i(x)>0\}$ all satisfy $x\in \mathrm{supp}{\varphi }_i\subseteq U_i$, so they form a simplex of $K$; hence the formula lands in $\mathrm{\mid }K\mathrm{\mid }$. Finite sums of continuous coordinates make $f$ continuous.

Let $\mathrm{sd}K$ be the barycentric subdivision and $h:\mathrm{\mid }\mathrm{sd}K\mathrm{\mid }\to \mathrm{\mid }K\mathrm{\mid }$ its canonical homeomorphism, sending the vertex indexed by a face $\sigma$ to the barycenter of $\sigma$. For each nonempty face $\sigma$, choose $x_{\sigma }\in U_{\sigma }$; there are only finitely many such choices. Define $g:\mathrm{\mid }\mathrm{sd}K\mathrm{\mid }\to F$ on vertices by $g(\sigma )=x_{\sigma }$. Inductively, a simplex of $\mathrm{sd}K$ is a chain ${\sigma }_0⊊\cdots ⊊{\sigma }_m$. Its face opposite ${\sigma }_0$ has already been mapped into $U_{{\sigma }_1}\subseteq U_{{\sigma }_0}$; cone that face to $x_{{\sigma }_0}$ along the unique minimizing geodesics in $U_{{\sigma }_0}$. This extension maps the entire chain simplex into $U_{{\sigma }_0}$. Deleting ${\sigma }_0$ gives the already assigned opposite face; deleting any other vertex restricts the same coning formula to the corresponding face. Thus all face definitions agree, and finite simplex-by-simplex pasting makes $g$ continuous.

For each $x\in F$, the support of $f(x)$ is a face $\sigma$ whose every $U_i$ contains $x$. Every chain simplex containing $h^{-1}f(x)$ has a smallest face ${\sigma }_0\subseteq \sigma$, so $x\in U_{{\sigma }_0}$; step 2.2 also gives $g{h}^{-1}f(x)\in U_{{\sigma }_0}$. The open set $P={\bigcup }_i(U_i\times U_i)$ contains every pair $(x,g{h}^{-1}f(x))$. On each $U_i\times U_i$, the unique minimizing connector is continuous by [F1]; connectors from different members agree on overlaps because they are globally unique. They therefore define a continuous connector on $P$. Applying it to $x$ and $g{h}^{-1}f(x)$ gives a homotopy $g{h}^{-1}f\simeq {\mathrm{id}}_F$.

For a face ${\sigma }_0$ of $K$, let $C_{{\sigma }_0}$ be the subcomplex of faces $\eta$ such that $\eta \cup {\sigma }_0$ is a face. It contracts to the barycenter of ${\sigma }_0$: for any point in a simplex $\eta$ of $C_{{\sigma }_0}$, the straight segment to that barycenter lies in the simplex $\eta \cup {\sigma }_0$. On a chain simplex with smallest face ${\sigma }_0$, the subdivision homeomorphism $h$ lands in $C_{{\sigma }_0}$; step 2.2 puts $g$ in $U_{{\sigma }_0}$, so every active vertex of $fg$ forms a face jointly with ${\sigma }_0$, and $fg$ also lands in $C_{{\sigma }_0}$. A face of the chain simplex has either the same smallest face or a larger one, whose carrier is contained in $C_{{\sigma }_0}$. Induct over the finite simplices of $\mathrm{sd}K$: the two endpoint maps and the already constructed boundary homotopy define a map from the boundary of each prism into its contractible carrier $C_{{\sigma }_0}$; coning that boundary map to the carrier's barycenter extends the homotopy over the prism, relative to its boundary. The finite extensions paste continuously and give $fg\simeq h$. Together with step 3.1, this shows $g{h}^{-1}$ is a homotopy inverse for $f$: the two composites are homotopic to ${\mathrm{id}}_F$ and ${\mathrm{id}}_{\mathrm{\mid }K\mathrm{\mid }}$, respectively.