Finite cw pairs admit finite simplicial homotopy models

We first describe a finite simplicial replacement for the cylinder of a simplicial map $f:P\to Q$. Start with $Q$ and an edge from each vertex $v$ of $P$ to $f(v)$, using disjoint domain vertices. Having constructed the part over $\mathrm{\partial }\sigma$ for a simplex $\sigma$ with image face $\tau$, adjoin the cone on $M(f{\mathrm{\mid }}_{\mathrm{\partial }\sigma }:\mathrm{\partial }\sigma \to \tau )$ with a fresh apex. This contains the subdivided simplex as the cone on its subdivided boundary. The base already retracts to the contractible simplex $\tau$, so both base and cone are contractible; the inclusion is a CW homotopy equivalence and F2 gives a strong deformation retraction to the base. Attach these finite pieces along their specified faces. Fresh vertices and the shared face construction make the intersections exactly subcomplexes in the sense of F3.

Successively retract the cone pieces in decreasing dimension. The resulting retraction $r:M(f)\to Q$ carries each domain simplex into its image face $\tau$, so $r{\mathrm{\mid }}_P$ is homotopic there to $f$ by straight lines. Extend that endpoint adjustment over $M(f)$, fixed on $Q$, by the CW HEP used in F2. Thus the domain inclusion is homotopic in $M(f)$ to $f$ with the prescribed endpoint. Adjoin a cone on the subdivided copy of $P$; the result $C(f)$ is a finite simplicial complex. This construction does not assume the ordinary mapping cylinder itself is simplicial.

Here is the induction with subpairs retained. Begin with the finitely many vertices of $A$, and adjoin its cells in increasing dimension. At each stage retain a common CW space $Z$ having both the current CW space and its simplicial model $Y$ as deformation retracts. Given finitely many new attaching maps $S^{r-1}\to X_{\mathrm{o}\mathrm{l}\mathrm{d}}$, retract to $Y$ and approximate simplicially by F1 after finitely many subdivisions of the sphere domains. For $r=0$, just adjoin isolated vertices. For $r\ge 1$, adjoin the complexes $M(f)$ to $Z$ along $Y$. Each added subdivided sphere is homotopic in this common space to its original attaching map, by the old retraction homotopy, the simplicial approximation homotopy, and the cylinder homotopy just built.

For each of these homotopies glue $D^r\times I$ along its boundary cylinder. The bottom contains the original attached cell, and the top contains the cone on the new sphere; the top simplicial space is $Y\cup C(f)$ over all the new cells. The prism retractions in F2 give deformation retractions to the two end adjunctions. The old retractions extend over an attaching disk by first extending the boundary homotopy over its collar; this is again the relative prism construction, fixed on the unaltered old end. Hence the new common space has both new ends as deformation retracts. This is a finite construction and preserves the earlier common subspace wherever no new cell is being attached.

After finishing $A$, keep its common space $Z_A$ and its two retracts $A$ and $L$. Now repeat the same construction for the cells of $X\setminus A$ in increasing dimension, with $Z_A$ designated as the subspace throughout. The homotopies on $Z_A$ are exactly its old retraction homotopies and remain within $Z_A$; every prism newly attached for a cell outside $A$ leaves this designated subspace alone. Thus the final common pair $(Z,Z_A)$ retracts as a pair to $(X,A)$ and to a finite simplicial pair $(K,L)$. Composing its retractions gives the required pair homotopy equivalence. Empty $A$ uses $Z_A=\mathrm{\varnothing }$, and empty $X$ gives the empty simplicial pair.