Finite classifying-approximation and coherence input

Write $U(r)=\{U\in M_r(\mathbb{C}):U^{\ast }U=I\}$ and $\mathfrak{u}(r)=\{X\in M_r(\mathbb{C}):X^{\ast }+X=0\}$. A smooth Hermitian rank-$r$ bundle is a smooth rank-$r$ complex vector bundle with a smoothly varying positive Hermitian inner product; its topological version uses a continuous bundle and continuous metric. A unitary connection is a complex-linear connection satisfying the Leibniz rule and $X⟨s,t⟩=⟨{\mathrm{\nabla }}_{X}s,t⟩+⟨s,{\mathrm{\nabla }}_{X}t⟩$ for smooth sections $s,t$ and vector fields $X$. Write ${\mathrm{\nabla }}_{X}s=(\mathrm{\nabla }s)(X)$; its curvature is the $\mathrm{End}(E)$-valued two-form $F_{\mathrm{\nabla }}(X,Y)=[{\mathrm{\nabla }}_X,{\mathrm{\nabla }}_Y]-{\mathrm{\nabla }}_{[X,Y]}$. A symmetric polynomial of degree $m$ is a symmetric $m$-linear form on $\mathfrak{u}(r)$, invariant here under simultaneous conjugation by $U(r)$; evaluation on curvature means $p_c(F_{\mathrm{\nabla }},\dots ,F_{\mathrm{\nabla }})$ using exterior products and the stated normalization. A unitary bundle isomorphism is a smooth complex-linear fiberwise isometry (or a continuous one for topological bundles). Let $BU(r)={\mathrm{G}\mathrm{r}}_r({\mathbb{C}}^{\mathrm{\infty }})$ be the direct limit of finite complex Grassmannians with its direct-limit CW topology, and let ${\gamma }_r\to BU(r)$ be the universal topological Hermitian rank-$r$ bundle. For a finite-dimensional Hausdorff second-countable smooth manifold $M$ without boundary of dimension $d$, integers $r,m\ge 1$, and a real-valued symmetric $U(r)$-invariant $m$-linear form $p_c:\mathfrak{u}(r{)}^m\to \mathbb{R}$ with a specified characteristic normalization, a finite classifying-approximation and coherence input consists of the following data. The normalization includes every required $2\pi i$ factor. For unitary connections ${\mathrm{\nabla }}_0,{\mathrm{\nabla }}_1$ on the same Hermitian bundle, use the affine path ${\mathrm{\nabla }}_t={\mathrm{\nabla }}_0+t({\mathrm{\nabla }}_1-{\mathrm{\nabla }}_0)$ and the transgression convention ${\mathrm{CS}}_{p_c}({\mathrm{\nabla }}_0,{\mathrm{\nabla }}_1)=m{\int }_0^1{p}_c({\mathrm{\nabla }}_1-{\mathrm{\nabla }}_0,F_{{\mathrm{\nabla }}_t},\dots ,F_{{\mathrm{\nabla }}_t})dt,$ which satisfies $d{\mathrm{CS}}_{p_c}=p_c(F_{{\mathrm{\nabla }}_1})-p_c(F_{{\mathrm{\nabla }}_0})$ by Explicit Chern–Simons transgression between two connections.

First, it includes a finite-dimensional smooth manifold $N$, a smooth Hermitian rank-$r$ bundle $F\to N$ with unitary connection ${\mathrm{\nabla }}_F$, and a continuous map $u:N\to BU(r)$ inducing isomorphisms on ${\pi }_j$ for $j\le d$ and a surjection on ${\pi }_{d+1}$ (with chosen basepoints). A specified unitary isomorphism of topological bundles $F\cong u^{\ast }{\gamma }_r$ makes $u$ a classifying map for $F$. For every supplied smooth Hermitian bundle $E\to M$ with unitary connection $\mathrm{\nabla }$, it includes a smooth map $f:M\to N$ and a unitary bundle isomorphism $E\cong f^{\ast }F$.

Second, for every two such presentations $(N_0,F_0,{\mathrm{\nabla }}_{F_0},u_0,f_0)$ and $(N_1,F_1,{\mathrm{\nabla }}_{F_1},u_1,f_1)$ of the same $(E,\mathrm{\nabla })$, it includes a finite-dimensional smooth common approximation $N^{\mathrm{\prime }}$ with a $(d+1)$-connected continuous classifying map $u^{\mathrm{\prime }}:N^{\mathrm{\prime }}\to BU(r)$, a smooth Hermitian bundle $F^{\mathrm{\prime }}\to N^{\mathrm{\prime }}$ whose underlying bundle is isomorphic to $u^{\mathrm{\prime }\ast }{\gamma }_r$, and a unitary connection ${\mathrm{\nabla }}^{\mathrm{\prime }}$ on $F^{\mathrm{\prime }}$. It includes smooth comparison maps $a_i:N_i\to N^{\mathrm{\prime }}$ and unitary bundle isomorphisms $F_i\cong a_i^{\ast }{F}^{\mathrm{\prime }}$. Put ${\mathrm{\nabla }}_i^E=(a_i\circ f_i{)}^{\ast }{\mathrm{\nabla }}^{\mathrm{\prime }}$ using these isomorphisms. The data further include a smooth homotopy $h:M\times [0,1]\to N^{\mathrm{\prime }}$ from $a_0{f}_0$ to $a_1{f}_1$ and a unitary isomorphism $h^{\ast }{F}^{\mathrm{\prime }}\cong {\mathrm{pr}}_M^{\ast }E$ whose endpoint identifications are the given presentations. Transport the pullback connection $h^{\ast }{\mathrm{\nabla }}^{\mathrm{\prime }}$ through this isomorphism to a connection $\tilde{\mathrm{\nabla }}$ on ${\mathrm{pr}}_M^{\ast }E$, and write ${\mathrm{\nabla }}_t$ for its restriction to $M\times \{t\}$. Put $T_h={\int }_0^1{p}_c(F_{\tilde{\mathrm{\nabla }}})$, contracting with ${\mathrm{\partial }}_t$ and integrating in the increasing $t$ direction. For every unitary connection $\mathrm{\nabla }$ on $E$, the supplied coherence form $\lambda \in {\mathrm{\Omega }}^{2m-2}(M)$ satisfies the exact equation $-T_h+{\mathrm{CS}}_{p_c}({\mathrm{\nabla }}_0^E,\mathrm{\nabla })-{\mathrm{CS}}_{p_c}({\mathrm{\nabla }}_1^E,\mathrm{\nabla })=d\lambda ,$ where ${\mathrm{CS}}_{p_c}$ is the normalized path transgression of Explicit Chern–Simons transgression between two connections. If $2m-1>d$, all forms of degree $2m-1$ on $M$ vanish, so this identity is automatic with zero left side.

Finally, each approximation used when $2m-1\le d$ satisfies $H^{2m-1}(N;\mathbb{R})=0$. When $2m-1>d$, the corresponding group on $M$ vanishes by dimension; no vanishing of $H^{2m-1}(N;\mathbb{R})$ is required. These are explicitly supplied inputs for the conditional construction and uniqueness argument; this definition does not assert that they exist for every $M,E$.

Smooth manifolds and their smooth charts

The Axiom of Choice

The existence of the finite approximations, the common-refinement system, the odd-degree cohomology vanishing and the coherent transgression comparison are not proved by this definition. The differential-refinement item remains escalated until these inputs are supplied with a local existence argument.