Detecting periodic elements in higher topological Hochschild homology

Veen, Torleif¹

¹ Department of Mathematics, University of Bergen, Bergen, Norway

Geometry & topology, Tome 22 (2018) no. 2, pp. 693-756.

Résumé

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.

Détail

DOI : 10.2140/gt.2018.22.693

Classification : 55P42, 55P91, 55T99
Keywords: THH, K-theory, spectral sequences, Morava K-theory

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