The cyclotomic trace and algebraic K-theory of spaces

The cyclotomic trace is a map from algebraic K-theory of a group ring to a certain topological refinement of cyclic homology. The target is naturally mapped to topological Hochschild homology, and the cyclotomic trace lifts the topological Dennis trace. Our cyclic homology can be defined also for "group rings up to homotopy", and in this setting the cyclotomic trace produces invariants of Waldhausen's A-theory. Our main applications go in two directions. We show on the one hand that the K-theory assembly map is rationally injective for a large class of discrete groups, including groups which have finitely generated Eilenberg-MacLane homology in each degree. This is the analogue in algebraic K-theory of Novikov's conjecture about homotopy invariance of higher signatures. It implies for Quillen's K-groups the inclusion

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