Deligne and Cheeger–Simons models agree under geometric cycle inputs

In the truncated integer-form model of [F1], let ${\mathcal{A}}_k$ be the subcomplex with ${\mathrm{\Omega }}^j$ in degree $j+1$ for $0\le j<k$ and zero elsewhere. The quotient is the constant integer sheaf in degree zero, so $0\to {\mathcal{A}}_k\to {\mathcal{D}}_k\to \underset{‾}{\mathbb{Z}}\to 0$ is a short exact sequence of complexes. The partition-of-unity contraction of the form rows, as in the proof of [F1]'s exact sequence, computes ${\mathbb{H}}^k({\mathcal{A}}_k)={\mathrm{\Omega }}^{k-1}\mathrm{/}d{\mathrm{\Omega }}^{k-2}$ and ${\mathbb{H}}^{k+1}({\mathcal{A}}_k)=0$, with ${\mathrm{\Omega }}^{-1}=0$. The resulting long exact sequence gives a form action $a_{\mathrm{a}\mathrm{l}\mathrm{l}}:{\mathrm{\Omega }}^{k-1}\to {\widehat{H}}_{\mathrm{D}\mathrm{e}\mathrm{l}}^k$ with image $\ker I$. Normalize this action by the cone cocycle $(0,d\eta ,-\eta )$. Under [F1]'s projection, the degree-$(k-1)$ form component is multiplied by $(-1{)}^{k-1}$, so $a_{\mathrm{a}\mathrm{l}\mathrm{l}}(\eta )$ is represented by $(-1{)}^k\eta$ in the global top-form term of the truncated integer-form complex. The cone differential is $(z,\alpha ,b)\mapsto (dz,d\alpha ,-db+z-\alpha )$; hence this cocycle has curvature $d\eta$ and agrees with [F1]'s $a$ on closed forms. If $a_{\mathrm{a}\mathrm{l}\mathrm{l}}(\eta )=0$, its curvature gives $d\eta =0$, and [F1]'s exact sequence then says $[\eta {]}_{\mathrm{d}\mathrm{R}}$ is the real image of an integral class. Conversely any such closed form acts trivially by that sequence. By de Rham evaluation these are precisely the closed integral-period forms. Thus $\ker a_{\mathrm{a}\mathrm{l}\mathrm{l}}={\mathrm{\Omega }}_{\mathbb{Z}}^{k-1}$. These constructions commute with restriction.

Let $z$ be a smooth integral $(k-1)$-cycle. By [F4], choose an open neighborhood $U$ of its compact image with $H^k(U;\mathbb{Z})=0$. Step 1.1 supplies a representative $a_{\mathrm{a}\mathrm{l}\mathrm{l}}(\eta )$ of the restricted Deligne class, for $\eta \in {\mathrm{\Omega }}^{k-1}(U)$; define $\chi (z)={\int }_z\eta (\mathrm{m}\mathrm{o}\mathrm{d}\mathbb{Z})$. Two choices differ by a closed integral-period form, so give the same value. Given a second neighborhood, [F4] supplies a smaller such neighborhood inside their intersection; the kernel calculation of step 1.1 there proves independence of the neighborhood as well.

A common neighborhood for the finite union of two cycle supports gives one form representative, so integration proves additivity in cycles and in the Deligne class. For a boundary $z=\mathrm{\partial }c$, use [F4]'s cycle-representative condition to write $z-P=\mathrm{\partial }b$ inside a neighborhood $U$ with $H^k(U;\mathbb{Z})=0$, where $P$ is the supplied pseudomanifold fundamental cycle. The same condition supplies $y$ with $\mathrm{\partial }y=P$ in a neighborhood $V$ with $H^k(V;\mathbb{Z})=0$. Choose representatives $\eta$ on $U$ and ${\eta }^{\mathrm{\prime }}$ on $V$ by step 1.1. On $U\cap V$ their form-action classes agree, so $\eta -{\eta }^{\mathrm{\prime }}$ has integral periods. Since $P$ lies in the intersection, ${\int }_P\eta ={\int }_P{\eta }^{\mathrm{\prime }}(\mathrm{m}\mathrm{o}\mathrm{d}\mathbb{Z})$. Stokes and $d\eta =d{\eta }^{\mathrm{\prime }}=R(\widehat{x})$ give $\chi (z)={\int }_{b}R(\widehat{x})+{\int }_{y}R(\widehat{x})(\mathrm{m}\mathrm{o}\mathrm{d}\mathbb{Z})$. Now $c-b-y$ is a smooth integral $k$-cycle, and [F1] implies that $R(\widehat{x})$ has integral periods. Therefore $\chi (\mathrm{\partial }c)={\int }_{c}R(\widehat{x})(\mathrm{m}\mathrm{o}\mathrm{d}\mathbb{Z})$, proving the character identity. For a smooth map, choose a form representative near the image cycle and pull it back near the original cycle; a smaller neighborhood as in step 2.1 proves equality of evaluations. Restriction and refinement naturality follow in the same way.

The construction commutes with the form-action map by its defining integral. Its characteristic class is the connecting class of the local integer Cech cocycle: the third condition in [F4] states the precise equality ${\mathrm{\Psi }}_{\mathrm{\infty },M,\mathcal{U}}^k(c_{\mathrm{D}\mathrm{e}\mathrm{l}})=[\mathrm{Int}(R)-\delta \tilde{\chi }]$ in $H^k(C_{\mathrm{\infty }}^{\ast }(M;\mathbb{Z}))$, with independence of the cochain lift supplied by [F2]. Thus the comparison commutes with $I$ as well as $R$ and the form action; naturality of restriction and refinement gives the same compatibility for pullbacks and cover refinements.

The induced map between the exact rows is the identity on the form quotient, closed integral forms, real cohomology and integral cohomology terms identified in [F1] and [F3]. By [F5] it is an isomorphism on the middle differential-cohomology group. The proof is conditional on [F4]; no claim is made here that every smooth manifold has that local geometric input.