#### 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
##### Abstract

Given a commutative ring spectrum $R$, let ${\Lambda }_{X}R$ be the Loday functor constructed by Brun, Carlson and Dundas. Given a prime $p\ge 5$, we calculate ${\pi }_{\ast }\left({\Lambda }_{{S}^{n}}H{\mathbb{F}}_{p}\right)$ and ${\pi }_{\ast }\left({\Lambda }_{{T}^{n}}H{\mathbb{F}}_{p}\right)$ for $n\le p$, and use these results to deduce that ${v}_{n-1}$ in the ${\left(n-1\right)}^{st}$ connective Morava K-theory of ${\left({\Lambda }_{{T}^{n}}H{\mathbb{F}}_{p}\right)}^{h{T}^{n}}$ 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 ${\pi }_{\ast }\left({\Lambda }_{{T}^{n}}H{\mathbb{F}}_{p}\right)$, 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 ${\pi }_{\ast }\left({\Lambda }_{{T}^{n}}H{\mathbb{F}}_{p}\right)$ and is a vital tool in the calculations above.

##### Keywords
THH, K-theory, spectral sequences, Morava K-theory
##### Mathematical Subject Classification 2010
Primary: 55P42, 55P91, 55T99