2020/05/09 by Campbell, Jonathan A., Lind, John A., Malkiewich, Cary +2
#18D05 #55N15 #55P42 #Algebraic Topology (math.AT) #Category Theory (math.CT) #FOS: Mathematics #K-Theory and Homology (math.KT)
paper · doi:10.48550/arxiv.2005.04334
We show that the characteristic polynomial and the Lefschetz zeta function are manifestations of the trace map from the K-theory of endomorphisms to topological restriction homology (TR). Along the way we generalize Lindenstrauss and McCarthy's map from K-theory of endomorphisms to topological restriction homology, defining it for any Waldhausen category with a compatible enrichment in orthogonal spectra. In particular, this extends their construction from rings to ring spectra. We also give a revisionist treatment of the original Dennis trace map from K-theory to topological Hochschild homology (THH) and explain its connection to traces in bicategories with shadow (also known as trace theories).