Character functor and Character Diagram

A degree-$k$ Character Functor, for $k\ge 1$, on the category of finite-dimensional Hausdorff second-countable smooth manifolds and smooth maps consists of a contravariant functor ${\widehat{G}}^k$ to abelian groups and natural maps $i_1:H^{k-1}(M;\mathbb{R}\mathrm{/}\mathbb{Z})\to {\widehat{G}}^k(M),i_2:{\mathrm{\Omega }}^{k-1}(M)\mathrm{/}{\mathrm{\Omega }}_{\mathbb{Z}}^{k-1}(M)\to {\widehat{G}}^k(M),R:{\widehat{G}}^k(M)\to {\mathrm{\Omega }}_{\mathbb{Z}}^k(M),I:{\widehat{G}}^k(M)\to H^k(M;\mathbb{Z})\mathrm{.}$ Here ${\mathrm{\Omega }}_{\mathbb{Z}}^j$ is the group of closed smooth $j$-forms with integral periods; $\alpha$ is coefficient reduction, $B$ the Bockstein, and $\rho :H^k(M;\mathbb{Z})\to H^k(M;\mathbb{R})$ coefficient extension. The map $\beta :H^{k-1}(M;\mathbb{R})\to {\mathrm{\Omega }}^{k-1}(M)\mathrm{/}{\mathrm{\Omega }}_{\mathbb{Z}}^{k-1}(M)$ sends a real cohomology class to any closed-form representative modulo integral-period forms; $d$ sends $[\eta ]$ to $d\eta$, and $s:{\mathrm{\Omega }}_{\mathbb{Z}}^k(M)\to H^k(M;\mathbb{R})$ is the de Rham class. Require both displayed sequences in the Definition to be exact and require $i_1\circ \alpha =i_2\circ \beta ,I\circ i_1=-B,R\circ i_2=d,\rho \circ I=s\circ R\mathrm{.}$ These identities, together with the two exact sequences, specify the commutative Character Diagram.

Exact sequences for ordinary differential cohomology

The two exact rows are $0⟶H^{k-1}(M;\mathbb{R}\mathrm{/}\mathbb{Z})\stackrel{i_1}{\to }{\widehat{G}}^k(M)\stackrel{R}{\to }{\mathrm{\Omega }}_{\mathbb{Z}}^k(M)⟶0$ and $0⟶{\mathrm{\Omega }}^{k-1}(M)\mathrm{/}{\mathrm{\Omega }}_{\mathbb{Z}}^{k-1}(M)\stackrel{i_2}{\to }{\widehat{G}}^k(M)\stackrel{I}{\to }{H}^k(M;\mathbb{Z})⟶0.$ The coefficient maps $\alpha ,B,\rho$ are those of $0\to \mathbb{Z}\to \mathbb{R}\to \mathbb{R}\mathrm{/}\mathbb{Z}\to 0$. The form maps $\beta ,d,s$ are defined in the Statement. A morphism of Character Functors is a natural transformation commuting with $i_1,i_2,R,I$ and acting as the identity on the adjacent coefficient and form functors.

Cheeger–Simons differential character

Exact sequences for ordinary differential cohomology

The first diagram displays the two exact rows. The following diagrams display the Bockstein, form-action, characteristic-class, curvature, and de Rham compatibility maps with the signs fixed in the Definition. They record algebraic data and are not proofs.