Categorification of the Farrell–Jones conjecture via hermitian K-theory
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IMT 1R2 207
Salle Pellos
The Farrell–Jones conjecture relates the K-theory and L-theory of a group ring R[G] to the K-theory and L-theory of R via the so-called assembly maps. I will recall consequences of the Farrell–Jones Conjecture in algebra and topology, most notably the Borel Conjecture about topological rigidity of aspherical manifolds. I will then present work by myself and others which revisits the Farrell–Jones conjecture by means of higher-categorical tools, mainly Efimov’s continuous K-theory on one hand and hermitian K-theory on the other. This results in a categorical model for the assembly maps. As an application, we recover Ranicki’s splitting of the L-theory of twisted Laurent polynomials. Finally, with an eye to the Geometric Langlands Program, I will report about work-in-progress on a version of the Farrell–Jones Conjecture for Hecke algebras of p-adic groups, following Bartels and Lück.
This is partially joint work with Jordan Levin and Victor Saunier.