2019/08/24 by Gabriel Angelini‐Knoll, Angelini-Knoll, Gabriel, J. D. Quigley +1
Mathematics · Computer Science · Physics and Astronomy · #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis #Black Holes and Theoretical Physics
paper · pdf · doi:10.48550/arxiv.1908.09164
The family of Thom spectra y(n) interpolates between the sphere spectrum and the mod two Eilenberg--MacLane spectrum. Computations of Mahowald, Ravenel, Shick, and the authors show that the associative ring spectrum y(n) has type n. Using trace methods, we give evidence that algebraic K-theory preserves this chromatic complexity. Our approach sheds light on the chromatic complexity of topological negative cyclic homology and topological periodic cyclic homology, which approximate algebraic K-theory and are of independent interest. Our main contribution is a homological approach that can be applied in great generality, such as to associative ring spectra R without additional structure whose coefficient rings are not completely understood.