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($\mathbb{Z}$).

中文翻译：

---

实拓扑霍克希尔德同调

本文从乘法范数构造的角度解释了 Hesselholt 和 Madsen 的实拓扑 Hochschild 同调函子 THR。我们证明 THR 满足共终性和 Morita 不变性，并且它是适当的乘法。然后我们计算它的几何不动点和它的分量的 Mackey 函子，并给出群代数的分解结果。使用这些结构结果，我们确定了 THR($\mathbb{F}_p$) 的同伦类型，并表明其双级同伦群在 $H\mathbb{F}_p$ 的双级同伦群上的一个生成器上是多项式的。然后我们计算远离素数$2$的THR($\mathbb{Z}$)的同伦类型，以及几何不动点谱的同伦环$\Phi^{\mathbb{Z}/2}$THR ($\mathbb{Z}$)。