Real topological Hochschild homology via the norm and Real Witt vectors

TL;DR

This paper characterizes Real topological Hochschild homology using the norm from cyclic groups to O(2), introduces a new algebraic analogue, and defines a novel theory of Witt vectors for rings with anti-involution.

Contribution

It provides a new characterization of Real THH via the norm, establishes a double coset formula, and introduces a new theory of Witt vectors for rings with anti-involution.

Findings

Explicit computation of homotopy groups for D_{2m}-Mackey functors.

A new multiplicative double coset formula for restrictions.

Definition of a new algebraic analogue of Real THH.

Abstract

We prove that Real topological Hochschild homology can be characterized as the norm from the cyclic group of order $2$ to the orthogonal group $O(2)$. From this perspective, we then prove a multiplicative double coset formula for the restriction of this norm to dihedral groups of order $2m$. This informs our new definition of Real Hochschild homology of rings with anti-involution, which we show is the algebraic analogue of Real topological Hochschild homology. Using extra structure on Real Hochschild homology, we define a new theory of $p$-typical Witt vectors of rings with anti-involution. We end with an explicit computation of the degree zero $D_{2m}$-Mackey functor homotopy groups of $\operatorname{THR}(\underline{\mathbb{Z}})$ for $m$ odd. This uses a Tambara reciprocity formula for sums for general finite groups, which may be of independent interest.