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<p>We show that the characteristic polynomial and the Lefschetz zeta function are manifestations of the trace map from the <i>K</i>-theory of endomorphisms to topological restriction homology (TR). Along the way we generalize Lindenstrauss and McCarthy’s map from <i>K</i>-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 <i>K</i>-theory to topological Hochschild homology (<InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44007_2025_154_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="38" /> </InlineMediaObject> <EquationSource Format="TEX">\({{\,\textrm{THH}\,}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mspace width="0.166667em" /> <mtext>THH</mtext> <mspace width="0.166667em" /> </mrow> </math></EquationSource> </InlineEquation>) and explain its connection to traces in bicategories with shadow (also known as trace theories).</p>

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

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年5月19日

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  • Jonathan A. Campbell,
  • John A. Lind,
  • Cary Malkiewich,
  • Kate Ponto,
  • Inna Zakharevich

摘要

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