Transfer maps and the cyclotomic trace

Abstract

We analyze the equivariant restriction (or transfer) maps in topological Hochschild homology associated to inclusions of group rings of the form R[H]→R[G], where R is a symmetric ring spectrum, G is a discrete group and H⊆G is a subgroup of finite index. This leads to a complete description of the associated restriction (or transfer) maps in topological cyclic homology in terms of the well-known stable transfers in equivariant stable homotopy theory. More generally, we analyze the restriction maps encountered in connection with monoid rings such as polynomial rings and truncated polynomial rings. As a first application of these results we prove a conjecture by Bökstedt, Hsiang and Madsen on how the transfer maps in Waldhausen's algebraic K-theory of spaces relate to the transfers in the stable equivariant homotopy category of a finite cyclic group. As a second application we calculate the subgroup of transfer invariant homotopy classes and we show that the TC-analogue of the lower K-groups vanish below degree −1.