Subdivision compatible continuous polyhedral homology comparison

Let $K^{\mathrm{\prime }}$ be a finite subdivision of $K$, with $L^{\mathrm{\prime }}$ the induced subdivision of $L$. The identity realization map is cellular from the old filtration to the new one, since $\mathrm{\mid }{K}^r\mathrm{\mid }\subset \mathrm{\mid }(K^{\mathrm{\prime }}{)}^r\mathrm{\mid }$. On an ordered old $r$-simplex, the sum of its new oriented $r$-simplices has all interior faces cancelled in pairs and has boundary the subdivided old boundary. Starting with vertices and using the boundary characterization in F1, it represents the old relative simplex class: the boundary map for the disk pair is injective, with reduced target for $r=1$. Thus the induced cellular map is the signed subdivision chain map, with coefficient $g$ unchanged.

The same argument applies to singular homology with $G$ coefficients. Hence the comparison square for the identity between the two triangulations commutes on their relative cell groups and on the skeletal lift isomorphisms. It follows that the homology comparison agrees before and after subdivision. Two successive subdivisions are covered by repetition; two finite linear subdivisions have a common refinement, obtained by triangulating their finite convex intersection cells in increasing face dimension. Applying the same argument to that refinement gives independence of its choice.

For a continuous map of finite pairs choose a simplicial approximation after a common barycentric subdivision of the source, by F2. F1 gives naturality for that simplicial map, the preceding step identifies the subdivided comparison with the original one, and homotopy invariance replaces the approximation by the given map in both theories. Thus the comparison is natural for the actual continuous map and cannot depend on the approximation chosen.

The pair-boundary square already commutes for the ordered-simplex comparison in F1. Subdivision and its comparison are maps of pairs and preserve the cone orientation, so they preserve that square. Therefore the resulting continuous natural comparison commutes with pair boundaries as claimed. Empty pairs and zero-dimensional triangulations use the same vertex/direct-sum comparison.