$K$-theory of endomorphisms, the $TR$-trace, and zeta functions

Abstract

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).

Minor revisions to make references consistent with arXiv:2006.04006. arXiv:2006.04006 is a companion to this paper and provides extended discussion of technical foundations

References in corpus (7)

• Homotopy limits and colimits and enriched homotopy theory
• Duality and traces for indexed monoidal categories
• Homotopical Intersection Theory, II: equivariance
• K-theory of endomorphisms via noncommutative motives
• The topological cyclic homology of the dual circle
• Facets of the Witt Vectors
• Spectral Waldhausen categories, the $S_{\bullet }$-construction, and the Dennis trace

Cited by in corpus (1)

• Spectral Waldhausen categories, the $S_{\bullet }$-construction, and the Dennis trace