Real topological Hochschild homology via the norm and Real Witt vectors

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 $\mathrm{THR}(\underline{\mathbb{Z}})$ for $m$ odd. This uses a Tambara reciprocity formula for sums for general finite groups, which may be of independent interest.

33 pages including references, 1 figure. Streamlined version. Published in Adv. Math

References in corpus (6)

• Equivariant symmetric monoidal structures
• Stable real K-theory and real topological Hochschild homology
• Real topological cyclic homology of spherical group rings
• On the geometric fixed-points of real topological cyclic homology
• Free incomplete Tambara functors are almost never flat
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Cited by in corpus (2)

• Real topological Hochschild homology of schemes
• Reflexive homology and involutive Hochschild homology as equivariant Loday constructions