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

/pdf/the-cyclotomic-trace-and-algebraic-k-theory-of-spaces-2p6y0dyn6s.pdf

Δ-sets i: homotopy theory

The cyclotomic trace and algebraic K-theory of spaces

Versions of the finiteness obstruction and simple homotopy theory “within e overX” are developed. This provides a setting for obstructions to the map analogs of the end ands-cobordism theorems for manifolds. These are applied to study equivariant mapping cylinder neighborhoods in topological group actions, triangulations of locally triangulable spaces, and block bundle structures on approximate fibrations.

Ends of maps, II

A Riemann–Roch theorem is a theorem which asserts that some algebraically defined wrong–way map in K –theory agrees or is compatible with a topologically defined one [BFM]. Bismut and Lott [BiLo] proved a Riemann–Roch theorem for smooth fiber bundles in which the topologically defined wrong–way map is the homotopy transfer of Becker–Gottlieb and Dold. We generalize and refine their theorem. In the process, we prove a family index theorem for fiber bundles with compact topological manifold fibers, a theorem in which the relevant index equation involves algebraic K –theory. Our methods enable us to make a “universal” choice of algebraic K –theory for such bundles. With this choice, we obtain index–theoretic characterizations of bundles of compact topological manifolds and bundles of compact smooth manifolds, respectively, among fibrations with finitely dominated fibers.

/pdf/a-parametrized-index-theorem-for-the-algebraic-k-theory-1hvtvmisxt.pdf

A parametrized index theorem for the algebraic K-theory Euler class

https://pure.mpg.de/pubman/item/item_3118526_1/component/file_3118527/Randal-Williams_Homological%20stability_oa_2017.pdf

Δ-sets i: homotopy theory

Citations

The cyclotomic trace and algebraic K-theory of spaces

Ends of maps, II

A parametrized index theorem for the algebraic K-theory Euler class

Homological stability for automorphism groups

The cyclic homology of an exact category

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