Algebraic $K$-theory, assembly maps, controlled algebra, and trace methods

Abstract

We give a concise introduction to the Farrell-Jones Conjecture in algebraic $K$-theory and to some of its applications. We survey the current status of the conjecture, and we illustrate the two main tools that are used to attack it: controlled algebra and trace methods.

Final version, to appear in Space - Time - Matter. Analytic and Geometric Structures (Jochen Brüning and Matthias Staudacher, eds.), De Gruyter. v3: updated references, added index, fixed few typos. 44 pages

References in corpus (3)

• The K-theoretic Farrell-Jones Conjecture for hyperbolic groups
• Algebraic K-theory of group rings and the cyclotomic trace map
• Splitting the relative assembly map, Nil-terms and involutions

Cited by in corpus (8)

• A vanishing theorem for tautological classes of aspherical manifolds
• Nilpotent invariance of semi-topological K-theory of dg-algebras and the lattice conjecture
• Poly-freeness of Artin groups and the Farrell-Jones Conjecture
• K-theory and actions on Euclidean retracts
• Equivariant Morse theory on Vietoris-Rips complexes & universal spaces for proper actions
• Assembly maps for topological cyclic homology of group algebras
• Two examples of vanishing and squeezing in $K_1$
• The Farrell-Jones Conjecture for normally poly-free groups