2022/08/22 by Land, Markus, Nikolaus, Thomas, Schlichting, Marco
#Algebraic Topology (math.AT) #FOS: Mathematics #K-Theory and Homology (math.KT) #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.2208.10556
We establish a formula for the L-theory spectrum of real C^*-algebras from which we deduce a presentation of the L-groups in terms of the topological K-groups, extending all previously known results of this kind. Along the way, we extend the integral comparison map τ\colon k → L obtained in previous work by the first two authors to real C^*-algebras and interpret it using topological Grothendieck-Witt theory. Finally, we use our results to give an integral comparison between the Baum-Connes conjecture and the L-theoretic Farrell-Jones conjecture, and discuss our comparison map τ in terms of the signature operator on oriented manifolds.