Algebraic K-theory of real topological K-theory

Gabriel Angelini-Knoll, Christian Ausoni, John Rognes

Published 2023-09-20Version 1

We calculate the A(1)-homotopy of the topological cyclic homology of the connective real topological K-theory spectrum ko, and show that it is a finitely generated and free F_2[v_2^{32}]-module of even rank between 390 and 444, on explicit generators in stems -1 \le *\le 198. This is achieved by using syntomic cohomology of ko as introduced by Hahn-Raksit-Wilson, extending work of Bhatt-Morrow-Scholze from the case of classical rings to E_\infty rings. In our case there are nontrivial differentials in the motivic spectral sequence from syntomic cohomology to topological cyclic homology, unlike in the case of complex K-theory at odd primes that was studied by Hahn-Raksit-Wilson.