Chern–Weil map for a chosen connection

Let $M$ be a finite-dimensional Hausdorff second-countable smooth manifold, possibly with boundary. Let $E\to M$ be a smooth rank-$r$ real or complex vector bundle with a supplied $G$-frame atlas, where $G$ is a real or complex matrix Lie group, and let $\mathrm{\nabla }$ be a fixed connection compatible with that reduction. For the algebra of finite sums of $\mathrm{Ad}(G)$-invariant polynomials $P={\sum }_k{P}_k$ with values in $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$. For each $P_k$, write $P_k^{\mathrm{p}\mathrm{o}\mathrm{l}}$ for its normalized symmetric polarization (with $P_0^{\mathrm{p}\mathrm{o}\mathrm{l}}=P_0$), and define ${\mathrm{CW}}_{E,\mathrm{\nabla }}(P)={\sum }_k[P_k^{\mathrm{p}\mathrm{o}\mathrm{l}}({\mathrm{\Omega }}_{\mathrm{\nabla }},\dots ,{\mathrm{\Omega }}_{\mathrm{\nabla }})]\in H_{\mathrm{d}\mathrm{R}}^{\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}}(M;\mathbb{K})\mathrm{.}$ where a degree-$k$ polynomial maps to cohomological degree $2k$. The map is a graded unital $\mathbb{K}$-algebra homomorphism, with polynomial multiplication on the source and wedge product on the target, and ${\mathrm{CW}}_{E,\mathrm{\nabla }}(1)=[1]$. The coefficient convention for $H_{\mathrm{d}\mathrm{R}}(M;\mathbb{K})$ on boundary manifolds is specified below. This definition depends on the fixed connection; independence of its cohomology value is a later theorem.

For the real coefficient target, use the ordinary real de Rham complex when $\mathrm{\partial }M=\mathrm{\varnothing }$ and the locally extendible boundary-chart complex from [F5] when $M$ has boundary; write its cohomology ring as $H_{\mathrm{d}\mathrm{R}}^{\bullet }(M;\mathbb{R})$. Its product is induced by wedge: the graded Leibniz rule in [F5] makes exact changes of a closed representative exact. For $\mathbb{K}=\mathbb{R}$, set $H_{\mathrm{d}\mathrm{R}}^{\bullet }(M;\mathbb{K})=H_{\mathrm{d}\mathrm{R}}^{\bullet }(M;\mathbb{R})$. For $\mathbb{K}=\mathbb{C}$, set ${\mathrm{\Omega }}^{\bullet }(M;\mathbb{C})={\mathrm{\Omega }}^{\bullet }(M;\mathbb{R}){\otimes }_{\mathbb{R}}\mathbb{C},d_{\mathbb{C}}=d_{\mathbb{R}}\otimes 1,H_{\mathrm{d}\mathrm{R}}^{\bullet }(M;\mathbb{C})=H^{\bullet }({\mathrm{\Omega }}^{\bullet }(M;\mathbb{C}),d_{\mathbb{C}})\mathrm{.}$ This is the complex-valued smooth-form complex, with boundary coefficients locally extendible componentwise. Real and imaginary parts split its cycles and exact forms, so its cohomology is $H_{\mathrm{d}\mathrm{R}}^{\bullet }(M;\mathbb{R}){\otimes }_{\mathbb{R}}\mathbb{C}$; wedge and the unit extend $\mathbb{C}$-linearly. On the empty manifold the target is the zero algebra with $1=0$, as in [F7].

For a polynomial $P_k$ homogeneous of degree $k$, use its polarization $P_k^{\mathrm{p}\mathrm{o}\mathrm{l}}$ in the global form from [F2]. Define the map by taking its cohomology class and summing over the homogeneous components. No connection-independence assertion is included in the definition.