Trace maps from the algebraic K-theory of the integers (after Marcel Bo¨kstedt)

Abstract

Let p be any prime. We consider Bo¨kstedt's topological refinement K(ℤ) → T(ℤ) = THH(ℤ)of the Dennis trace map from algebraic K-theory of the integers to topological Hochschild homology of the integers. This trace map is shown to induce a surjection on homotopy in degree2 p − 1, onto the first P-torsion in the target. Furthermore, Bo¨kstedt's map factors through the S 1-homotopy fixed points T(ℤ) hS 1 of T(ℤ), and it is shown that the first p-torsion element in degree2 p − 3 of the stable homotopy groups of spheres is detected in the homotopy of T(ℤ) hS 1 . Both results are due to Bo¨kstedt, but have remained unpublished.