Real topological Hochschild homology

TL;DR

This paper provides a comprehensive analysis of real topological Hochschild homology (THR), including structural properties, calculations of homotopy types for specific rings, and decomposition results, advancing understanding of THR's algebraic and topological features.

Contribution

It introduces new structural results for THR, computes its homotopy types for _p and , and explores its multiplicative and fixed point properties, extending previous work in the field.

Findings

THR satisfies cofinality and Morita invariance.

Homotopy type of THR(_p) is determined and shown to be polynomial.

Homotopy type of THR() away from 2 is calculated.

Abstract

This paper interprets Hesselholt and Madsen's real topological Hochschild homology functor THR in terms of the multiplicative norm construction. We show that THR satisfies cofinality and Morita invariance, and that it is suitably multiplicative. We then calculate its geometric fixed points and its Mackey functor of components, and show a decomposition result for group-algebras. Using these structural results we determine the homotopy type of THR($\mathbb{F}_p$) and show that its bigraded homotopy groups are polynomial on one generator over the bigraded homotopy groups of $H\mathbb{F}_p$. We then calculate the homotopy type of THR($\mathbb{Z}$) away from the prime $2$, and the homotopy ring of the geometric fixed-points spectrum $\Phi^{\mathbb{Z}/2}$THR($\mathbb{Z}$).