Good-cover Cech-to-singular comparison input

For this page, a good cover of a finite-dimensional Hausdorff second-countable smooth manifold $M$ means an ordered locally finite open cover $\mathcal{U}=(U_i{)}_{i\in I}$ such that every nonempty finite intersection admits a smooth contraction to a point, together with a chosen locally finite smooth partition of unity subordinate to $\mathcal{U}$. A good-cover Cech-to-singular comparison input for $(M,\mathcal{U})$ consists of natural isomorphisms ${\check{H}}^q(\mathcal{U};\underset{‾}{\mathbb{Z}})\cong H_{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}^q(M;\mathbb{Z})$ for all $q\ge 0$, compatible with cover refinements, restrictions to open subsets, and smooth pullbacks with compatible cover functions. Here $\underset{‾}{\mathbb{Z}}$ is the sheaf of locally constant integer-valued functions and the left side is fixed-cover Cech cohomology. For use in Deligne hypercohomology comparison, the supplied input also requires, on each nonempty finite intersection $U_J$, the explicit acyclicity conditions $H^q(U_J,\underset{‾}{\mathbb{Z}})=0$ and $H^q(U_J,{\mathrm{\Omega }}^j)=0$ for every $q>0$ and every smooth-form degree $j\ge 0$. These are hypotheses of that use; they are not inferred here from contractibility alone. The Cech-to-singular isomorphisms themselves are proved by Natural integral good-cover Cech to singular comparison.

Fixed-cover Čech cohomology

Singular cohomology with coefficients

The canonical maps and their existence for every stated good cover are proved in Natural integral good-cover Cech to singular comparison. The proof uses integral small singular chains and the Cech–singular double complex, so torsion is retained. Its maps commute with refinements, open restrictions, smooth pullbacks and coefficient homomorphisms. This item still defines the supplied comparison data; the proof is in that separate lemma, with its explicit full-AC convention.