Homology operations in the topological cyclic homology of a point

Abstract

We consider the commutative $\mathbb{S}$–algebra given by the topological cyclic homology of a point. The induced Dyer–Lashof operations in mod $p$ homology are shown to be nontrivial for $p=2$, and an explicit formula is given. As a part of the calculation, we are led to compare the fixed point spectrum ${\mathbb{S}}^G$ of the sphere spectrum and the algebraic $K$–theory spectrum of finite $G$–sets, as structured ring spectra.