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).
| Original language | English |
|---|---|
| Pages (from-to) | 214-292 |
| Number of pages | 79 |
| Journal | Matematica |
| Volume | 4 |
| Issue number | 2 |
| DOIs | |
| State | Published - Jun 2025 |
Bibliographical note
Publisher Copyright:© The Author(s) 2025.
Funding
JC would like to thank Andrew Blumberg, Mike Mandell, and Randy McCarthy for helpful conversations about this paper, and for general wisdom about trace methods. CM would like to thank Randy McCarthy for persistently telling him about the TR trace for years \u2013 it\u2019s beginning to sink in a little. KP was partially supported by NSF grant DMS-1810779 and the University of Kentucky Royster Research Professorship. IZ was supported in part by NSF CAREER-1846767. The authors thank Cornell University for hosting the initial meeting which led to this work.
| Funders | Funder number |
|---|---|
| Andrew Blumberg | |
| Mike Mandell | |
| Cornell High Energy Synchrotron Source, Cornell University | |
| U.S. Department of Energy Chinese Academy of Sciences Guangzhou Municipal Science and Technology Project Oak Ridge National Laboratory Extreme Science and Engineering Discovery Environment National Science Foundation National Energy Research Scientific Computing Center National Natural Science Foundation of China | DMS-1810779 |
| U.S. Department of Energy Chinese Academy of Sciences Guangzhou Municipal Science and Technology Project Oak Ridge National Laboratory Extreme Science and Engineering Discovery Environment National Science Foundation National Energy Research Scientific Computing Center National Natural Science Foundation of China | |
| University of Kentucky Royster | CAREER-1846767 |
Keywords
- Algebraic K-theory
- Bicategories with shadow
- Dennis trace
- Topological Hochschild homology
- Topological restriction homology
ASJC Scopus subject areas
- General Mathematics