Differential extension of a generalized cohomology theory

Let $\mathsf{M}\mathsf{f}\mathsf{d}$ be the category of finite-dimensional Hausdorff second-countable smooth manifolds without boundary and smooth maps. Let $E$ be a commutative ring spectrum, meaning a spectrum with a unital, homotopy-associative and homotopy-commutative multiplication and its coherence data; for $M\in \mathsf{M}\mathsf{f}\mathsf{d}$, write $M_{+}$ for $M$ with a disjoint basepoint and define $E^n(M):=[{\mathrm{\Sigma }}^{\mathrm{\infty }}{M}_{+},{\mathrm{\Sigma }}^{n}E]$ in the stable homotopy category. Let $V={⨁}_{r\in \mathbb{Z}}{V}^r$ be a graded real vector space and define ${\mathrm{\Omega }}^m(M;V)={⨁}_{p+r=m}{\mathrm{\Omega }}^p(M;\mathbb{R}){\otimes }_{\mathbb{R}}{V}^r$, with total differential $d=d_{\mathrm{d}\mathrm{R}}\otimes 1$. Write $H_{\mathrm{d}\mathrm{R}}^m(M;V)$ for the degree-$m$ cohomology of this complex, and fix a natural degree-preserving transformation $\mathrm{ch}:E^{\ast }(M)\to H_{\mathrm{d}\mathrm{R}}^{\ast }(M;V)$. A differential extension of this data consists of graded contravariant groups ${\widehat{E}}^n(M)$ and natural transformations

$I:{\widehat{E}}^n(M)\to E^n(M),R:{\widehat{E}}^n(M)\to {\mathrm{\Omega }}_{\mathrm{c}\mathrm{l}}^n(M;V),a:{\mathrm{\Omega }}^{n-1}(M;V)\mathrm{/}d{\mathrm{\Omega }}^{n-2}(M;V)\to {\widehat{E}}^n(M),$

where these maps are natural under smooth pullback and ${\mathrm{\Omega }}_{\mathrm{c}\mathrm{l}}^n=\ker (d:{\mathrm{\Omega }}^n(M;V)\to {\mathrm{\Omega }}^{n+1}(M;V))$. Require $[R(x){]}_{\mathrm{d}\mathrm{R}}=\mathrm{ch}(I(x))$, $I(a([\eta ]))=0$, and $R(a([\eta ]))=d\eta$. Here $[\eta ]$ is the quotient class modulo exact forms. Define ${\mathrm{ch}}_{\mathrm{\Omega }}:E^{n-1}(M)\to {\mathrm{\Omega }}^{n-1}(M;V)\mathrm{/}d{\mathrm{\Omega }}^{n-2}(M;V)$ by taking the unique quotient class of closed forms corresponding to $\mathrm{ch}$ under de Rham theory. Require the sequence

$E^{n-1}(M)\stackrel{{\mathrm{ch}}_{\mathrm{\Omega }}}{\to }{\mathrm{\Omega }}^{n-1}(M;V)\mathrm{/}d{\mathrm{\Omega }}^{n-2}(M;V)\stackrel{a}{\to }{\widehat{E}}^n(M)\stackrel{I}{\to }{E}^n(M)\to 0$

to be exact. Finally require the homotopy formula $f_1^{\ast }x-f_0^{\ast }x=a({\int }_{[0,1]}{F}^{\ast }R(x))$ for every pair $M,N\in \mathsf{M}\mathsf{f}\mathsf{d}$, every smooth homotopy $F:M\times [0,1]\to N$ from $f_0$ to $f_1$, and every $x\in {\widehat{E}}^n(N)$, using the orientation from $0$ to $1$. A differential extension is not assumed unique, and multiplicativity of its differential groups is additional structure rather than part of this definition.

Sequential prespectra, spectra, and adjoint structure maps

Exact sequences for ordinary differential cohomology

Complex K-theory is a two-periodic generalized cohomology theory

The structure maps and their compatibility are represented by the commutative square below. The lower map sends a closed form to its de Rham cohomology class; the square alone asserts compatibility, not exactness of any sequence.

The ordinary differential-cohomology supplier fixes a familiar special case of this interface. The complex $K$-theory supplier fixes one possible topological theory and Bott grading used by the later cycle model. Neither supplier implies existence or uniqueness of a differential extension for an arbitrary $E$.