Character-diagram uniqueness for ordinary differential cohomology

Let $M\in {\mathcal{C}}_{\mathrm{g}\mathrm{e}\mathrm{o}}$, $g\in {\widehat{G}}^k(M)$, and $a\in Z_{k-1}^{\mathrm{\infty }}(M;\mathbb{Z})$. By [F4], choose an open neighborhood $U$ of $\mathrm{\mid }a\mathrm{\mid }$ with $H^k(U;\mathbb{Z})=0$. Exactness of the second row in [F1] then gives a form class $[\theta ]\in {\mathrm{\Omega }}^{k-1}(U)\mathrm{/}{\mathrm{\Omega }}_{\mathbb{Z}}^{k-1}(U)$ whose form-action image is the restriction of $g$. Define $\mathrm{\Phi }(g)(a)={\int }_a\theta (\mathrm{m}\mathrm{o}\mathrm{d}\mathbb{Z})$. Changing $\theta$ by a closed integral-period form does not change the value. If $U^{\mathrm{\prime }}$ is another choice, a common smaller neighborhood from [F4] and naturality of the restriction maps show that both definitions restrict to the same form class and give the same integral. Thus $\mathrm{\Phi }(g)(a)$ is well-defined.

To see additivity in the cycle, apply [F4] to the finite union of supports of $a,b,a+b$ and use one common neighborhood $U$. There $\mathrm{\Phi }(g)(a+b)={\int }_{a+b}\theta ={\int }_a\theta +{\int }_b\theta =\mathrm{\Phi }(g)(a)+\mathrm{\Phi }(g)(b)$. Additivity in $g$ follows because the exact-row form-action map is a homomorphism and the same local representatives add.

Let $a=\mathrm{\partial }e$ be a boundary, with $e$ a smooth integral $k$-chain. By [F4], replace $a$ by the fundamental cycle $P$ of an oriented piecewise-smooth pseudomanifold, with $a-P=\mathrm{\partial }b$ in a neighborhood $U$ on which $g=i_2[\theta ]$; this local form exists because $H^k(U;\mathbb{Z})=0$ and the second exact row of [F1] is onto. The same input supplies a chain $y$ with $\mathrm{\partial }y=P$ in a neighborhood $U^{\mathrm{\prime }}$ on which $g=i_2[{\theta }^{\mathrm{\prime }}]$. [F7] gives ${\int }_a\theta -{\int }_P\theta ={\int }_{b}R(g)$ and ${\int }_P{\theta }^{\mathrm{\prime }}={\int }_{y}R(g)$. On $U\cap U^{\mathrm{\prime }}$, injectivity of $i_2$ in [F1] implies $\theta -{\theta }^{\mathrm{\prime }}$ has integral periods, so its integral on $P$ is an integer. Since $\mathrm{\partial }(e-b-y)=0$, the integral of $R(g)$ on $e-b-y$ is an integer by $R(g)\in {\mathrm{\Omega }}_{\mathbb{Z}}^k$ in [F1]. Combining these equalities gives $\mathrm{\Phi }(g)(a)={\int }_{e}R(g)(\mathrm{m}\mathrm{o}\mathrm{d}\mathbb{Z})$. Thus $\mathrm{\Phi }(g)$ is a differential character with curvature $R(g)$.

Let $u\in H^{k-1}(M;\mathbb{R}\mathrm{/}\mathbb{Z})$ and let $a$ be a smooth integral cycle. Choose $U$ from [F4]. Since $H^k(U;\mathbb{Z})=0$, the Bockstein of $u{\mathrm{\mid }}_U$ vanishes and the coefficient long exact sequence gives a real class reducing to $u{\mathrm{\mid }}_U$. Choose a closed de Rham representative $\eta$ of that real class. Commutativity $i_1\alpha =i_2\beta$ in [F1] then gives $i_1(u){\mathrm{\mid }}_U=i_2([\eta ])$, where $[\eta ]$ here denotes the form class modulo integral-period forms. The definition of $\mathrm{\Phi }$ yields $\mathrm{\Phi }(i_1(u))(a)={\int }_a\eta (\mathrm{m}\mathrm{o}\mathrm{d}\mathbb{Z})$, which equals $u([a])$ by the evaluation identification [F6]; smooth-chain comparison [F2] makes this the same homology class used by $i_1$. Hence $\mathrm{\Phi }$ is the identity on the flat subgroup. On $i_2$, the same definition is integration of the supplied form; step 2.2 gives compatibility with $R$. The construction commutes with smooth maps because cycles and local forms pull back, and step 1.1 removes neighborhood choices. Thus $\mathrm{\Phi }$ is a natural morphism of the first exact rows in [F1] and [F3], with identity maps on $H^{k-1}(M;\mathbb{R}\mathrm{/}\mathbb{Z})$ and ${\mathrm{\Omega }}_{\mathbb{Z}}^k(M)$.

Apply [F5] to the two first exact rows in [F1] and [F3]. Step 3.1 gives identity maps on the outer terms, so the middle comparison $\mathrm{\Phi }$ is an isomorphism.

Now define $\tau :H^k(-;\mathbb{Z})\to H^k(-;\mathbb{Z})$ by $\tau (I_{\widehat{G}}(g))=I_{\widehat{H}}(\mathrm{\Phi }(g))$. Exactness and $i_2$ compatibility make this well-defined, and applying the same construction to ${\mathrm{\Phi }}^{-1}$ shows it is a natural automorphism. Every torsion class is a Bockstein $t=Bu$ because real cohomology has no torsion and the coefficient sequence is exact. Since $I{i}_1=-B$, represent $t$ by $i_1(-u)$. Step 3.1 gives $\mathrm{\Phi }(i_1(-u))=i_1(-u)$, so $\tau (t)=I_{\widehat{H}}(i_1(-u))=Bu=t$. The rigidity condition in [F4] forces $\tau =\mathrm{i}\mathrm{d}$, proving compatibility with $I$ and hence with the full Character Diagram.

If ${\mathrm{\Phi }}^{\mathrm{\prime }}$ is another natural morphism fixing the diagram entries, then for any cycle $a$ choose a neighborhood $U$ as in [F4]; the restriction of a class is a form-action class there, and both natural transformations give its same integral over $a$. Therefore ${\mathrm{\Phi }}^{\mathrm{\prime }}(g)(a)=\mathrm{\Phi }(g)(a)$ for every cycle, so ${\mathrm{\Phi }}^{\mathrm{\prime }}=\mathrm{\Phi }$. This proves uniqueness conditional on the geometric input in the Statement.