Character-diagram geometric input conditions

Let ${\mathcal{C}}_{\mathrm{g}\mathrm{e}\mathrm{o}}$ be the full subcategory of finite-dimensional Hausdorff second-countable smooth manifolds whose objects satisfy the following conditions in the stated smooth-singular convention. First, for every integer $r\ge 0$, every finite smooth singular $r$-chain $c$, and every open neighborhood $V$ of its compact image, there is an open $U$ with $\mathrm{\mid }c\mathrm{\mid }\subseteq U\subseteq V$ and $H^j(U;\mathbb{Z})=0$ for every $j>r$. Second, for every integer $r\ge 1$, every smooth singular $(r-1)$-cycle is homologous in each neighborhood of its image to the fundamental cycle of an oriented piecewise-smooth embedded $(r-1)$-pseudomanifold; if the cycle is null-homologous in $M$, such a representative bounds inside a neighborhood whose degree-$r$ integral cohomology vanishes. Here a piecewise-smooth pseudomanifold is a finite oriented union of smooth singular simplices for which every codimension-one face occurs twice with opposite induced orientations.

Third, fix $k\ge 1$, a good cover $\mathcal{U}$, and a degree-$k$ class $x\in {\widehat{H}}_{\mathrm{D}\mathrm{e}\mathrm{l}}^k(M;\mathbb{Z})$ represented by a total cocycle in the Cech complex of the Deligne model from Smooth Deligne cohomology model. Write $R(x)\in {\mathrm{\Omega }}_{\mathrm{c}\mathrm{l}}^k(M)$ for its curvature projection and $c_{\mathrm{D}\mathrm{e}\mathrm{l}}(x)\in {\check{H}}^k(\mathcal{U};\underset{‾}{\mathbb{Z}})$ for the Cech class of its integer characteristic projection, using the model's exponential-sequence connecting map. For a smooth integral singular chain $c$, put $\mathrm{Int}(R(x))(c)={\int }_{c}R(x)$; its cochain differential is positive precomposition with the boundary. The local input supplies, for each smooth integral $(k-1)$-cycle $z$, a neighborhood $U$ of $\mathrm{\mid }z\mathrm{\mid }$ with $H^k(U;\mathbb{Z})=0$ and a form class $[\eta ]\in {\mathrm{\Omega }}^{k-1}(U)\mathrm{/}{\mathrm{\Omega }}_{\mathbb{Z}}^{k-1}(U)$ whose Deligne form-action image is $x{\mathrm{\mid }}_U$, and requires that the formula $h_x(z)={\int }_z\eta (\mathrm{m}\mathrm{o}\mathrm{d}\mathbb{Z})$ be independent of all such choices and define a homomorphism $h_x:Z_{k-1}^{\mathrm{\infty }}(M;\mathbb{Z})\to \mathbb{R}\mathrm{/}\mathbb{Z}$ satisfying $h_x(\mathrm{\partial }c)={\int }_{c}R(x)(\mathrm{m}\mathrm{o}\mathrm{d}\mathbb{Z})$ for every smooth integral $k$-chain $c$. It further supplies a real cochain ${\tilde{h}}_x\in C_{\mathrm{\infty }}^{k-1}(M;\mathbb{R})=\mathrm{Hom}(C_{k-1}^{\mathrm{\infty }}(M;\mathbb{Z}),\mathbb{R})$ whose restriction to cycles is a real lift of $h_x$. Require the typed equality ${\mathrm{\Psi }}_{\mathrm{\infty },M,\mathcal{U}}^k(c_{\mathrm{D}\mathrm{e}\mathrm{l}}(x))=[\mathrm{Int}(R(x))-\delta {\tilde{h}}_x]\in H^k(C_{\mathrm{\infty }}^{\ast }(M;\mathbb{Z})),$ where $C_{\mathrm{\infty }}^{\ast }(M;\mathbb{Z})=\mathrm{Hom}(C_{\ast }^{\mathrm{\infty }}(M;\mathbb{Z}),\mathbb{Z})$ has differential by positive precomposition with $\mathrm{\partial }$ and ${\mathrm{\Psi }}_{\mathrm{\infty },M,\mathcal{U}}^k:=H^k({\iota }^{\ast })\circ {\mathrm{\Psi }}_{M,\mathcal{U},\mathbb{Z}}^k:{\check{H}}^k(\mathcal{U};\underset{‾}{\mathbb{Z}})⟶H^k(C_{\mathrm{\infty }}^{\ast }(M;\mathbb{Z})),$ with ${\mathrm{\Psi }}_{M,\mathcal{U},\mathbb{Z}}^k$ the natural integral good-cover comparison of Natural integral good-cover Cech to singular comparison, $\iota :C_{\ast }^{\mathrm{\infty }}(M;\mathbb{Z})\hookrightarrow C_{\ast }(M;\mathbb{Z})$ the chain inclusion, and ${\iota }^{\ast }$ restriction of cochains. The smooth-to-ordinary homology isomorphism of Smooth integral chains compute singular homology and the natural free-complex UCT of Singular UCT extension from cycle projections make $H^k({\iota }^{\ast })$ an isomorphism under full AC. Require these supplied cycle evaluations and cochain lifts to be natural under smooth pullbacks with compatible cover functions and under cover refinements.

Require the neighborhood objects supplied by the first two conditions to lie in ${\mathcal{C}}_{\mathrm{g}\mathrm{e}\mathrm{o}}$ and the category to be closed under their open restrictions. The subcategory is said to satisfy the character-diagram geometric input conditions when, in addition, every natural automorphism of the integral cohomology functor restricted to ${\mathcal{C}}_{\mathrm{g}\mathrm{e}\mathrm{o}}$ that fixes torsion classes is the identity. These conditions are supplied hypotheses; this definition does not assert that any specified nonempty class of manifolds satisfies them.

Smooth integral singular chain complex

Singular cohomology with coefficients

This item names the geometric, naturality, and Cech-to-chain inputs used in the local reconstruction of Character Diagram uniqueness and the Deligne comparison. It does not assert that the subcategory contains every smooth manifold or that its objects satisfy the named conditions. Simons–Sullivan cite dimension theory, triangulation and pseudomanifold perturbation arguments; the full local proofs, including the Cech-to-chain integer-cocycle comparison and natural cohomology rigidity, remain an owner escalation.