Detecting periodic elements in higher topological Hochschild homology

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.