Twisted differential generalized cohomology theories and their Atiyah-Hirzebruch spectral sequence
type: Item
instance of: scholarly article
title: Twisted differential generalized cohomology theories and their Atiyah-Hirzebruch spectral sequence
zbMATH Open document ID: 1427.19006
author: Hisham Sati
author: Daniel Joseph Grady
published in: Algebraic \& Geometric Topology
publication date: 2019-12-12T00:00:00Z
full work available at URL: https://arxiv.org/abs/1711.06650
review text: 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 \(\mat…
reviewed by: Do Ngoc Diep
Mathematics Subject Classification ID: 19L50
Mathematics Subject Classification ID: 53C05
Mathematics Subject Classification ID: 55S05
Mathematics Subject Classification ID: 55R20
Mathematics Subject Classification ID: 55T25
Mathematics Subject Classification ID: 14A20
Mathematics Subject Classification ID: 19K35
zbMATH DE Number: 7142622
zbMATH Keywords: differential topology
zbMATH Keywords: twisted cohomology
zbMATH Keywords: twisted spectra
zbMATH Keywords: twisted \(K\)-theory
zbMATH Keywords: Aiyah-Hirzebruch spectral sequence
MaRDI profile type: Publication
OpenAlex ID: W3098753275
arXiv ID: 1711.06650
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DOI: 10.2140/AGT.2019.19.2899
description: scientific article