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 $R[H]\to R[G]$, where $R$ is a symmetric ring spectrum, $G$ is a discrete group and $H\subseteq G$ is a subgroup of finite index. This leads to a complete description of the associated restriction (or transfer) maps in topological cyclic homology $$ \Res_G^H\co\TC(R[G])\to \TC(R[H]) $$ 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 B\"okstedt, 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 $$ \pi_*\TC(R[z_1,z_1^{-1},\dots,z_m,z_m^{-1}])^{\INV} $$