K-Theory of Endomorphisms, the TR-Trace, and Zeta Functions

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 ( $${{\,\textrm{THH}\,}}$$ ) and explain its connection to traces in bicategories with shadow (also known as trace theories).