Detecting periodic elements in higher topological Hochschild homology

Given a commutative ring spectrum $R$ , let $Λ_{X} R$ be the Loday functor constructed by Brun, Carlson and Dundas. Given a prime $p \geq 5$ , we calculate $π_{*} (Λ_{S^{n}} H F_{p})$ and $π_{*} (Λ_{T^{n}} H F_{p})$ for $n \leq p$ , and use these results to deduce that $v_{n - 1}$ in the ${(n - 1)}^{s t}$ connective Morava K-theory of ${(Λ_{T^{n}} H F_{p})}^{h T^{n}}$ is nonzero and detected in the homotopy fixed-point spectral sequence by an explicit element, whose class we name the Rognes class.

To facilitate these calculations, we introduce multifold Hopf algebras. Each axis circle in $T^{n}$ gives rise to a Hopf algebra structure on $π_{*} (Λ_{T^{n}} H F_{p})$ , and the way these Hopf algebra structures interact is encoded with a multifold Hopf algebra structure. This structure puts several restrictions on the possible algebra structures on $π_{*} (Λ_{T^{n}} H F_{p})$ and is a vital tool in the calculations above.

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Keywords

THH, K-theory, spectral sequences, Morava K-theory

Mathematical Subject Classification 2010

Primary: 55P42, 55P91, 55T99

References

Publication

Received: 6 March 2014

Revised: 2 May 2017

Accepted: 4 June 2017

Published: 16 January 2018

Proposed: Mark Behrens

Seconded: Paul Goerss, Haynes R Miller

Authors

Torleif Veen
Department of Mathematics University of Bergen Bergen Norway

Volume 22, issue 2 (2018)

Torleif Veen

Abstract