Volume 22, issue 2 (2018)

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Detecting periodic elements in higher topological Hochschild homology

Torleif Veen

Geometry & Topology 22 (2018) 693–756

Given a commutative ring spectrum R, let ΛXR be the Loday functor constructed by Brun, Carlson and Dundas. Given a prime p 5, we calculate π(ΛSnHFp) and π(ΛTnHFp) for n p, and use these results to deduce that vn1 in the (n1)st connective Morava K-theory of (ΛTnHFp)hTn 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 Tn gives rise to a Hopf algebra structure on π(ΛTnHFp), 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(ΛTnHFp) and is a vital tool in the calculations above.

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THH, K-theory, spectral sequences, Morava K-theory
Mathematical Subject Classification 2010
Primary: 55P42, 55P91, 55T99
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
Torleif Veen
Department of Mathematics
University of Bergen