Twisted differential generalized cohomology theories and their Atiyah-Hirzebruch spectral sequence
The paper is devoted to a construction of some Atiyah-Hirzebruch spectral sequences (AHSS) as the same ones for the twisted differential spectra as bundles of spectra equipped with a flat connection. Let \(M,N\) be smooth manifolds, \(\mathcal M f/M\) the \(\infty\)-overcategory on arrows \(N \to M\), with Grothendieck topology of good open covers, \(\mathcal Sp\) the category of spectra, \(\mathcal R\) the ordinary ring spectrum, \(\mathrm{const}(\mathcal R) : \mathcal Mf \to \mathcal Sp\) -- the prestack with Grothendieck topology of good open covers, \(\underline{\mathcal R}\) -- its stackification on \(M\) and the same over any locally constant sheaf of spectra on \(\mathcal Mf\). Let \(\mathrm{Pic}^{\mathrm{top}}_{\underline{\mathcal R}}(M)\) be the \(\infty\)-groupoid of invertible objects in the \(\infty\)-category of locally constant sheaves of \(\underline{\mathcal R}\)-module spectra over \(M\), with the properties of descent, correspondence with topological twist, twisted cohomology theory. The \textit{category of differential ring spectra} \(\mathrm{diff}(\mathcal Ring \mathcal Sp)\) appeared as the \((\infty,1)\)-pullback of composition \(H\circ \imath\) of localization \(\imath\) at weak equivalences and functors \(H\) of the Eilenberg-Mac Lane equivalence from the ordinary 1-category \(\mathbb R-\mathrm{CDGA}\) of commutative differential graded algebras to the \((\infty,1)\)-category \(\mathbb R-\mathcal Alg\mathcal Sp\) of commutative \(H\mathbb R\)-algebra spectra, and the smash product with Eilenberg-MacLane spectrum functor \(\wedge H\mathbb R\) from the \((\infty,1)\)-category of \(E_\infty\)-ring spectra to the \((\infty,1)\)-category \(\mathbb R-\mathcal Alg\mathcal Sp\) (Definittion 1). The de Rham complex with values in a CDGA is defined as \(\Omega(-,A) := \Omega^* \otimes_\mathbb R \underline{A}\) as the tensor product of the de Rham complex with the local constant sheaf of \(A\). A \textit{smooth stacks twist} is defined as a sheaf \(\mathcal L\) of \(\Omega(-,A)\)-modules on the restricted site \(\mathcal Mf/M\) which is invertible, K-flat and weak locally constant over \(\Omega^*(-,A)\) (Definition 2) and the triple \((\mathcal R_\tau, t, \mathcal L)\) is a \textit{differential twist}. (Defintion 3). The differential twists have the well properties like local triviality (Proposition 4) and the twisted differential cohomologies \(\hat{R}^*(-;\hat \tau)\) have good properties like additivity, local triviality, Meyer-Vietoris sequences like bundles. The canonical bundle of spectra over the Picard 1-groupoid which comes from the 1-Grothendieck construction. The local-global construction of twisted differential cohomology is done (Theorem 8) and the twisted differentials \(\hat {\mathcal R}_{\hat \tau}= ({\mathcal R}_\tau, t, \mathcal L)\) appeared as flat bundles \(E_\tau \to M\) of spectra with connection \(\nabla\) as the differential on \(\mathcal L\) of sections of the bundle \(E\wedge \wedge^\bullet(T^*M) \to M\). The ordinary techniques of reduction of the structure group and construction from local data of a flat bundles of spectra are applicable. The general construciton of Atiyah-Hirzebruch spectral sequence \(\widehat{\mathrm{AHSS}}_{\hat \tau}\) (Theorem 18) is done with \(E_2\)-page, based on \(\check C\)ech cohomology. The usual properties of \(\widehat{\mathrm{AHSS}}_{\hat \tau}\) like: linearity, normalization, naturality, module differential (Proposition 19). The AHSS in twisted differential K-theory looks like very good one related with the differential K-theory twisted Chern character \(\hat{\mathrm ch}_{\hat h}: \hat{\mathcal K}_{\hat h} \to \hat H \mathbb Q[u,u^{-1}]_{\hat h}\) over \(M\) (Definition 23). The obstructions associated to curvature forms are considered (Lemma 24, Proposition 26). It has a correspondence with Massey products (Proposition 27, Lemma 28). Various example are considered in \S3.5 for 3-sphere \(\mathbb S^3\). The paper is a good introduction to the subject.
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