Rational cohomology scalar comparison on compact smooth manifolds

By [F1] there are a finite CW complex $K$ and maps $h:X\to K$, $g:K\to X$ with $g\circ h$ homotopic to ${\mathrm{i}\mathrm{d}}_X$ and $h\circ g$ homotopic to ${\mathrm{i}\mathrm{d}}_K$. Homotopy invariance [F2] gives $g_{\ast }{h}_{\ast }=(g\circ h{)}_{\ast }=\mathrm{i}\mathrm{d}$ and $h_{\ast }{g}_{\ast }=(h\circ g{)}_{\ast }=\mathrm{i}\mathrm{d}$ on singular homology with any coefficients, so $h_{\ast }:H_n(X;\kappa )\to H_n(K;\kappa )$ is an isomorphism for every $n$ and every coefficient field $\kappa$. For a field $\kappa$ each cellular chain group $C_n^{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}(K;\kappa )$ is a direct sum of one copy of $\kappa$ per $n$-cell [F3], and the finite CW complex $K$ has finitely many cells, so $C_n^{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}(K;\kappa )$ is a finite-dimensional $\kappa$-vector space and its subquotient $H_n^{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}(K;\kappa )=\ker d_n\mathrm{/}\mathrm{im}{d}_{n+1}$ is finite-dimensional as well [F3]; by cellular homology equals singular homology [F3], $H_n(K;\kappa )$ and hence $H_n(X;\kappa )$ are finite-dimensional. In particular $H_n(X;\mathbb{Q})$ and $H_n(X;\mathbb{R})$ are finite-dimensional for every $n$.

Let $k$ be a field, let $F$ be a $k$-vector space, and let $D_{\ast }$ be a chain complex of $k$-vector spaces with differential $\mathrm{\partial }$, cycles $Z_n=\ker {\mathrm{\partial }}_n$ and boundaries $B_n=\mathrm{im}{\mathrm{\partial }}_{n+1}$. Under AC choose $k$-linear complements $A_n,T_n$ with $Z_n=B_n\oplus A_n$ and $D_n=Z_n\oplus T_n$; then ${\mathrm{\partial }}_n$ maps $T_n$ isomorphically onto $B_{n-1}$, because it is injective ($T_n\cap Z_n=0$) and ${\mathrm{\partial }}_n(D_n)=B_{n-1}={\mathrm{\partial }}_n(T_n)$. Tensoring these supplied splittings preserves them: tensor the inclusions and projections, whose composites are identities or zero and whose sum is the identity. Thus $D_n{\otimes }_{k}F=(B_n{\otimes }_{k}F)\oplus (A_n{\otimes }_{k}F)\oplus (T_n{\otimes }_{k}F)$, and the boundary is zero on the first two summands and an isomorphism from the third onto $B_{n-1}{\otimes }_{k}F$. Consequently $\ker ({\mathrm{\partial }}_n\otimes 1)=Z_n{\otimes }_{k}F$, $\mathrm{im}({\mathrm{\partial }}_{n+1}\otimes 1)=B_n{\otimes }_{k}F$, and $H_n(D_{\ast }{\otimes }_{k}F)\cong A_n{\otimes }_{k}F\cong H_n(D_{\ast }){\otimes }_{k}F$, the isomorphism sending the class of $z\otimes \mu$ to $[z]\otimes \mu$. For $D_{\ast }=C_{\ast }(X;\mathbb{Q})$ and $k=\mathbb{Q}$ the chain-level identification $C_{\ast }(X;\mathbb{Q}){\otimes }_{\mathbb{Q}}F\cong C_{\ast }(X;F)$ is the associativity and unit isomorphism for $C_n(X;\mathbb{Z}){\otimes }_{\mathbb{Z}}\mathbb{Q}{\otimes }_{\mathbb{Q}}F\cong C_n(X;\mathbb{Z}){\otimes }_{\mathbb{Z}}F$ [F4], so $H_n(X;\mathbb{Q}){\otimes }_{\mathbb{Q}}F\cong H_n(X;F)$ canonically by the map induced by $c\mapsto c\otimes 1$.

Fix a characteristic-zero field $F$ and a degree $n$, and put $V=H_n(X;\mathbb{Q})$, finite-dimensional by step 1.1. Evaluation gives natural isomorphisms $H^n(X;\kappa )\cong {\mathrm{Hom}}_{\kappa }(H_n(X;\kappa ),\kappa )$ for $\kappa =\mathbb{Q}$ and $\kappa =F$ [F5]; choosing a finite $\mathbb{Q}$-basis $(v_i)$ of the finite-dimensional space $V$ with dual basis $(f_i)$, the map $f_i\otimes \mu \mapsto (v_j\mapsto {\delta }_{ij}\mu )$ identifies ${\mathrm{Hom}}_{\mathbb{Q}}(V,\mathbb{Q}){\otimes }_{\mathbb{Q}}F$ with ${\mathrm{Hom}}_{\mathbb{Q}}(V,F)$, and $\psi \mapsto (v\otimes \mu \mapsto \psi (v)\mu )$ identifies ${\mathrm{Hom}}_{\mathbb{Q}}(V,F)$ with ${\mathrm{Hom}}_F(V{\otimes }_{\mathbb{Q}}F,F)$, with inverse $\chi \mapsto (v\mapsto \chi (v\otimes 1))$. Composing these isomorphisms with the identification $V{\otimes }_{\mathbb{Q}}F\cong H_n(X;F)$ of step 1.2 and the two evaluation isomorphisms gives a natural isomorphism $H^n(X;\mathbb{Q}){\otimes }_{\mathbb{Q}}F\cong H^n(X;F)$; on a cocycle $\phi$ and a chain $z\otimes \mu$ the image of $[\phi ]\otimes 1$ evaluates to $\phi (z)\mu$, exactly as the coefficient-change cochain $c\otimes \mu \mapsto \phi (c)\mu$ does, so the isomorphism is the natural coefficient map $[\phi ]\otimes 1\mapsto [\phi \otimes 1]$. Hence $H^n(X;\mathbb{Q}){\otimes }_{\mathbb{Q}}F\cong H^n(X;F)$ for every characteristic-zero field $F$, and $H^n(X;F)$ is finite-dimensional because $H_n(X;F)\cong V{\otimes }_{\mathbb{Q}}F$ is.

By [F6] the cup product on singular cochains with coefficients in a commutative unital ring is given by the front/back formula, and the coefficient-change map $\phi \mapsto \phi \otimes 1$ satisfies $((\phi ⌣\psi )\otimes 1)(\sigma )=(\phi \otimes 1)(\sigma [0,\dots ,p])\cdot (\psi \otimes 1)(\sigma [p,\dots ,p+q])$ on every singular simplex by multiplying the two coefficient values inside $F$, with $1\otimes 1=1$; hence coefficient change is a unital homomorphism of graded rings, and the isomorphism of step 2.1, being its $F$-linear extension, satisfies ${\mathrm{\Phi }}_F((a⌣b)\otimes \mu )={\mathrm{\Phi }}_F(a\otimes 1)⌣{\mathrm{\Phi }}_F(b\otimes \mu )$ for cohomology classes $a,b$ and scalars $\mu$, so the comparison isomorphisms preserve cup products. For a map $f:X\to Y$ of compact smooth manifolds the cochain identity $f^{\ast }(\phi \otimes 1)=(f^{\ast }\phi )\otimes 1$ holds because both sides send a simplex $\sigma$ to $\phi (f\circ \sigma )$, and pullback is a unital graded-ring homomorphism [F6]; hence the isomorphisms of step 2.1 commute with pullback and are natural for such maps, and multiplicativity is preserved as well.

The de Rham theorem [F7] gives the natural isomorphism of graded real algebras $H_{\mathrm{d}\mathrm{R}}^{\ast }(X)\cong H_{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}^{\ast }(X;\mathbb{R})$, so composing with step 2.1 for $F=\mathbb{R}$ identifies $H^n(X;\mathbb{Q}){\otimes }_{\mathbb{Q}}\mathbb{R}$ with real de Rham cohomology; the real de Rham theorem assumes ${\mathrm{A}\mathrm{C}}_{\omega }$, which AC supplies through [F1]. For $F=\mathbb{C}$ the complex de Rham complex is ${\mathrm{\Omega }}_{\mathbb{R}}^{\ast }(X){\otimes }_{\mathbb{R}}\mathbb{C}$ with differential $d\otimes 1$ [F7], and the argument of step 1.2 applied degreewise to this complex over the field extension $\mathbb{R}\subseteq \mathbb{C}$ gives $H_{\mathrm{d}\mathrm{R}}^n(X;\mathbb{C})\cong H_{\mathrm{d}\mathrm{R}}^n(X;\mathbb{R}){\otimes }_{\mathbb{R}}\mathbb{C}$; likewise $H^n(X;\mathbb{C})\cong H^n(X;\mathbb{R}){\otimes }_{\mathbb{R}}\mathbb{C}$, either by the same base change and duality [F5] or by associativity from $H^n(X;\mathbb{Q}){\otimes }_{\mathbb{Q}}\mathbb{C}\cong H^n(X;\mathbb{Q}){\otimes }_{\mathbb{Q}}\mathbb{R}{\otimes }_{\mathbb{R}}\mathbb{C}$ via step 2.1 for $F=\mathbb{R}$ and $F=\mathbb{C}$. Hence the complex de Rham comparison is the complexification of the real one, and the identification with real or complex de Rham cohomology holds for $F=\mathbb{R},\mathbb{C}$ as asserted, with naturality and cup-product compatibility from steps 2.1 and 3.1.