2023/07/27 by Sanjana Agarwal, Agarwal, Sanjana
Mathematics · Physics and Astronomy · #Homotopy and Cohomology in Algebraic Topology #Black Holes and Theoretical Physics #Algebraic structures and combinatorial models
paper · pdf · doi:10.48550/arxiv.2307.15135
The Morava E-theories, En, are complex-oriented 2-periodic ring spectra, with homotopy groups \mathbbW_\mathbbFpn[[u1, u2, ... , un-1]][u,u-1]. Here \mathbbW denotes the Witt vector ring. En is a Landweber exact spectrum and hence uniquely determined by this ring as BP∗-algebra. Algebraic K-theory of En is a key ingredient towards analyzing the layers in the p-complete Waldhausen K-theory chromatic tower. One hopes to use the machinery of trace methods to get results towards K-theory once the computation for THH(En) is known. In this paper we describe THH(E2) as part of consecutive chain of cofiber sequences where each cofiber sits in the next cofiber sequence and the first term of each cofiber sequence is describable completely in terms of suspensions and localizations of E2. For these results, we first calculate K(i)-homology of THH(E2) using a Bökstedt spectral sequence and then lift the generating classes of K(1)-homology to fundamental classes in homotopy group of THH(E2). These lifts allow us to construct terms of the cofiber sequence and explicitly understand how they map to THH(E2).