-theory of endomorphisms, the -trace, and zeta functions
arXiv:2005.04334
Abstract
We show that the characteristic polynomial and the Lefschetz zeta function are manifestations of the trace map from the -theory of endomorphisms to topological restriction homology (TR). Along the way we generalize Lindenstrauss and McCarthy's map from -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 -theory to topological Hochschild homology (THH) and explain its connection to traces in bicategories with shadow (also known as trace theories).
arXiv commentMinor 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 (18)
- Equivalences of monoidal model categories
- Localization theorems in topological Hochschild homology and topological cyclic homology
- On the non-existence of elements of Kervaire invariant one
- Homotopy limits and colimits and enriched homotopy theory
- Duality and traces for indexed monoidal categories
- Fixed point theory and trace for bicategories
- Localization for THH(ku) and the topological Hochschild and cyclic homology of Waldhausen categories
- On the Taylor Tower of Relative K-theory
- Traces in symmetric monoidal categories
- Periodic points and topological restriction homology
- K-theory of endomorphisms via noncommutative motives
- Homotopical Intersection Theory, II: equivariance
- Trace theories, Bokstedt periodicity and Bott periodicity
- The topological cyclic homology of the dual circle
- Facets of the Witt Vectors
- Parametrized spectra, a low-tech approach
- Coherence for indexed symmetric monoidal categories
- Spectral Waldhausen categories, the -construction, and the Dennis trace