Topological cyclic homology of the integers at two

Abstract

The topological Hochschild homology of the integers T( Z) = THH( Z) is an S 1- equivariant spectrum. We prove by computation that for the restricted C 2- action on T( Z) the fixed points and homotopy fixed points are equivalent, after passing to connective covers and completing at two. By Tsalidis (1994) a similar statement then holds for the action of every cyclic subgroup C 2 n ⊂ S 1 of order a power of two. Next we inductively determine the mod two homotopy groups of all the fixed point spectra T( Z) C 2 n , following Bökstedt and Madsen (1994, 1995) and Tsalidis (1994). We also compute the restriction maps relating these spectra, and use this to find the mod two homotopy groups of the topological cyclic homology of the integers TC( Z), and of the algebraic K- theory of the two-adic integers K( Z ̂ 2) .