2017/11/17 by Daniel Grady, Hisham Sati · 7 citations
Computer Science · Mathematics · Physics and Astronomy · #Algebra over a field #Cohomology #Construct (python library) #De Rham cohomology #Differential (mechanical device) #Differential form #Homotopy and Cohomology in Algebraic Topology #Nonlinear Waves and Solitons #Polynomial and algebraic computation #Sequence (biology) #Spectral sequence #math.AT #math.DG #math.KT
paper · pdf · doi:10.2140/agt.2019.19.2899
published in Algebraic & Geometric Topology 19(6), 2899-2960 (Mathematical Sciences Publishers) · 47 pages
arxiv created 2017/11/17 · openalex created_date 2017/12/04 · openalex publication_date 2019/10/20 · arxiv updated 2019/10/30 · openalex updated_date 2026/08/05
We construct the Atiyah–Hirzebruch spectral sequence (AHSS) for twisted differential generalized cohomology theories. This generalizes to the twisted setting the authors’ corresponding earlier construction for differential cohomology theories, as well as to the differential setting the AHSS for twisted generalized cohomology theories, including that of twisted [math] –theory by Rosenberg and by Atiyah and Segal. In describing twisted differential spectra we build on the work of Bunke and Nikolaus, but we find it useful for our purposes to take an approach that highlights direct analogies with classical bundles and that is at the same time amenable for calculations. We will, in particular, establish that twisted differential spectra are bundles of spectra equipped with a flat connection. Our prominent case will be twisted differential [math] –theory, for which we work out the differentials in detail. This involves differential refinements of primary and secondary cohomology operations the authors developed in earlier papers. We illustrate our constructions and computational tools with examples.