Natural integral good-cover Cech to singular comparison

Put $A^q={\mathrm{Hom}}_{\mathbb{Z}}(A_q,G)$ and $C^q={\mathrm{Hom}}_{\mathbb{Z}}(C_q(X;\mathbb{Z}),G)$. Restriction $\rho ={\iota }^{\ast }:C^{\ast }\to A^{\ast }$ is a cochain homotopy equivalence. Indeed $r^{\ast }$ is a cochain map, $\rho r^{\ast }=1$, and $r^{\ast }\rho -1=-d{D}^{\ast }-D^{\ast }d$, where $(D^{\ast }\phi )(a)=\phi (Da)$. These equations follow by precomposing [F2]'s integral chain identities and use precisely the positive convention [F3]. Thus $H^q(\rho )$ is invertible for every abelian $G$, without assuming that $\mathrm{Hom}(-,G)$ preserves arbitrary quasi-isomorphisms. This restriction map is canonical, although the homotopy inverse used to prove it is an isomorphism need not be.

For $p,q\ge 0$ define $K^{p,q}={\prod }_{i_0<\cdots <i_p}{C}^q(U_{i_0}\cap \cdots \cap U_{i_p};G)\mathrm{.}$ Use alternating extension in the cover indices. Let ${\delta }_h$ be alternating deletion followed by restriction and let $d_v$ be the singular coboundary on each intersection. Both square to zero and commute, because restriction of a cochain commutes with precomposition by the singular boundary. The total complex is ${\mathrm{Tot}}^{n}K={⨁}_{p+q=n}{K}^{p,q}$, with total differential $d_T={\delta }_h+(-1{)}^p{d}_v$ on bidegree $(p,q)$; the sum has finitely many bidegrees for each $n$. Define $a:A^{\ast }\to \mathrm{Tot}K$ by restricting a small cochain to each $U_i$, placing it in $K^{0,\ast }$. Its horizontal differential is zero on overlaps, and its vertical differential is the singular one, so it is a cochain map. Augment each horizontal row by $A^q$ in column $p=-1$. Given a small simplex $\sigma$, let $I_{\sigma }=\{i:\mathrm{im}\sigma \subset U_i\}$ and choose $i(\sigma )\in I_{\sigma }$ using [A1]. On a component evaluated at a singular simplex with image in $U_{i_0}\cap \cdots \cap U_{i_{p-1}}$, set $(hc{)}_{i_0\cdots i_{p-1}}(\sigma )=c_{i(\sigma ),i_0,\dots ,i_{p-1}}(\sigma )\mathrm{.}$ For $p=0$ this means $(hc)(\sigma )=c_{i(\sigma )}(\sigma )\in A^q$. Alternating deletion gives ${\delta }_{h}h+h{\delta }_h=1$ on the augmented row: the deletion of the newly prepended index gives $c$ and the other deletion terms cancel in pairs with opposite signs. Repetitions give zero in exactly the alternating convention. Thus every augmented row is split exact. No commutation between $h$ and $d_v$ is asserted or needed.

Let $B^p=C^p(\mathcal{U};\underset{‾}{G})$. Define $w:B^{\ast }\to \mathrm{Tot}K$ in bidegree $(p,0)$ by evaluating the locally constant function on each singular vertex in its intersection. A locally constant function is constant along a path, so its singular coboundary on every singular edge is zero. Thus $w$ commutes with the total differential and is a cochain map for any open cover. Now assume every nonempty finite intersection $U_J$ is contractible. A supplied contraction, its point inclusion and the constant map give integral chain maps whose composites are chain homotopic to the identity by the signed prism formula [F3]. Dualizing these homotopies into $G$ shows that $C^{\ast }(U_J;G)$ has the cohomology of the singular cochain complex of a point. That point complex has $G$ in each degree, with coboundary zero in even degrees and the identity in odd degrees: its degree-$q$ boundary sum is ${\sum }_{j=0}^{q+1}(-1{)}^j$. Hence its cohomology is $G$ in degree zero and zero in positive degrees. Since a nonempty contractible space is path connected, evaluation of a locally constant function is precisely its degree-zero cocycle identification. Consequently each column augmented by $B^p$ at $q=-1$ is exact; for products over intersections, choose all contractions simultaneously by [A1], or solve each cocycle component with their prism homotopies. This proves product exactness without a field or finite-cover hypothesis.

This augmented row exactness proves that $a$ is a quasi-isomorphism by a finite elimination, as follows. Add column $p=-1$ with entries $A^q$, horizontal map the augmentation $a$, and vertical part $-d$ in the total differential; the resulting total complex is the cochain mapping cone of $a$, with its degree-$n$ group ${\mathrm{Tot}}^{n}K\oplus A^{n+1}$ and differential $(z,u)\mapsto (d_{T}z+au,-du)$. Let $z$ be a cocycle of this augmented total complex. It has finitely many bidegrees $p+q=n$, $p\ge -1,q\ge 0$. Take its largest occupied $p$. In the cocycle equation at $(p+1,q)$, no vertical term from column $p+1$ is present, so ${\delta }_h{z}^{p,q}=0$. If $p\ge 0$, augmented row exactness gives $y^{p-1,q}$ with ${\delta }_h{y}^{p-1,q}=z^{p,q}$. Subtracting $d_{T}y$ removes that component and introduces only one component with smaller $p$. Iterating terminates at column $-1$; there, injectivity of the row augmentation makes a horizontally closed component zero. Hence every cocycle is a coboundary, also in degree $-1$, and the cone is acyclic. Equivalently $a$ induces isomorphisms: this equivalence can be seen directly from the cone equations, since an $a$-image that is a coboundary gives a cone cocycle, and a cocycle of the target lifts modulo a cone boundary. Therefore $H(a)$ is invertible in every degree.

Exactness of the augmented columns proves that $w$ is a quasi-isomorphism by the other finite elimination. Place $B^p$ in bidegree $(p,-1)$, use its Cech differential horizontally and its evaluation augmentation vertically, and retain $d_T={\delta }_h+(-1{)}^p{d}_v$. This augmented total complex is the cone of $w$ after multiplying its bottom-row coordinate in column $p$ by $(-1{)}^p$; under that identification its differential is $(z,u)\mapsto (d_{T}z+wu,-{\delta }_{h}u)$. For a cocycle, choose the largest occupied $q$. In the equation at $(p,q+1)$, no horizontal contribution from row $q+1$ is present, so $d_v{z}^{p,q}=0$. Column exactness supplies a preimage in $(p,q-1)$ if $q\ge 0$; multiply it by $(-1{)}^p$ before subtracting its total differential. This removes the component and introduces only a component with smaller $q$. The process terminates at $q=-1$, where injectivity of the evaluation augmentation makes the final component zero. There are finitely many bidegrees on each total diagonal, so no convergence, completion or spectral-sequence limiting argument occurs. Thus the cone is acyclic and $H(w)$ is an isomorphism.

Define, for every ordered open cover, ${\mathrm{\Psi }}_{X,\mathcal{U},G}^q=H^q(\rho {)}^{-1}{H}^q(a{)}^{-1}{H}^q(w)\mathrm{.}$ Steps 1.1 and 2.1 give the two inverse maps in this formula for any cover. Step 2.2 says that $H(w)$ is also invertible for a contractible-intersection cover, proving the asserted isomorphism. All three maps $\rho ,a,w$ were defined without choices; the cover-index selections and intersection contractions only proved their required properties. In degree zero, a Cech cocycle consists of compatible locally constant functions, hence glues to a function on $X$. Evaluating at vertices describes both $a\rho$ of that function and $w$ of its compatible family. The formula for ${\mathrm{\Psi }}^0$ therefore gives that same function, with no sign change. For $X=\mathrm{\varnothing }$ all groups are zero, and the conclusion still holds. The one-member cover of a nonempty contractible space has Cech cohomology $G$ in degree zero and zero otherwise, proving the stated singular-cohomology consequence.

For the pair $(f,c)$ in the Statement, postcomposition sends a $\mathcal{V}$-small singular simplex in $Y$ to a $\mathcal{U}$-small simplex in $X$, and commutes with all singular face maps. Precomposition therefore defines pullback maps on the full and small cochain complexes, commuting with $\rho$. On the double complex use $((f,c{)}^{\ast }k{)}_{j_0\cdots j_p}(\sigma )=k_{c(j_0),\dots ,c(j_p)}(f\circ \sigma )\mathrm{.}$ Every map is defined on the required intersection because $f(V_j)\subset U_{c(j)}$. Repeated indices are assigned zero and arbitrary reordering uses its sign. Alternating deletion shows this map commutes with ${\delta }_h$ term by term; postcomposition commuting with faces shows it commutes with $d_v$. It also commutes with $a$ and $w$, since both sides restrict/evaluate the same cochain or locally constant function on the same simplex. These three commuting squares, after taking cohomology and inverting $H(\rho )$ and $H(a)$, give exactly the stated naturality equality. Applying the same construction to a homomorphism $G\to G^{\mathrm{\prime }}$ gives coefficient naturality. In particular the inclusion $\mathbb{Z}\to \mathbb{R}$ is the literal coefficient change on singular and Cech cochains, not a tensor-duality identification.

A refinement is the case $f=\mathrm{id}$; if both covers have contractible intersections, naturality gives $H(c^{\mathrm{♯}})={\mathrm{\Psi }}_{\mathcal{V}}^{-1}{\mathrm{\Psi }}_{\mathcal{U}}$, independently of the refinement function. Taking $f:O\hookrightarrow X$ and the cover $(U_i\cap O)$ gives the open-restriction square without requiring that the restricted cover be good. For a supplied contractible-intersection cover of $O$ mapping to $\mathcal{U}$, the same formula gives the square between comparison isomorphisms. For arbitrary good-cover presentations of $X,Y$ and an arbitrary continuous $f:Y\to X$, define their cohomology pullback to be ${\mathrm{\Psi }}_{Y,\mathcal{V},G}^{-1}{f}^{\ast }{\mathrm{\Psi }}_{X,\mathcal{U},G}$. The identity and composition laws follow by cancelling adjacent comparison inverses and using the identity and composition laws for singular cochain pullback. The cover-map equality of step 4.1 shows that this agrees with the usual Cech pullback whenever an index map exists. It also makes identity maps between any two good-cover presentations canonical and transitive. Thus arbitrary open restriction or smooth pullback is available between any supplied good-cover presentations, without assuming that restricted intersections remain contractible or that a common good refinement has been supplied. Thus the theorem supplies the integral natural good-cover comparison interface under its exact cochain conventions, while making no false assertion that open restriction preserves good covers.