# Real topological Hochschild homology

### Emanuele Dotto

University of Warwick, Coventry, UK### Kristian Moi

KTH Royal Institute of Technology, Stockholm, Sweden### Irakli Patchkoria

University of Aberdeen, UK### Sune Precht Reeh

Copenhagen, Denmark

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## 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 its geometric fixed-points spectrum $\Phi^{\mathbb Z/2}$THR($\mathbb Z$).

## Cite this article

Emanuele Dotto, Kristian Moi, Irakli Patchkoria, Sune Precht Reeh, Real topological Hochschild homology. J. Eur. Math. Soc. 23 (2020), no. 1, pp. 63–152

DOI 10.4171/JEMS/1007