Vol. 7, No. 1, 2022

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$K$-theoretic torsion and the zeta function

John R. Klein and Cary Malkiewich

Vol. 7 (2022), No. 1, 77–118
Abstract

We generalize to higher algebraic K-theory an identity (originally due to Milnor) that relates the Reidemeister torsion of an infinite cyclic cover to its Lefschetz zeta function. Our identity involves a higher torsion invariant, the endomorphism torsion, of a parametrized family of endomorphisms as well as a higher zeta function of such a family. We also exhibit several examples of families of endomorphisms having nontrivial endomorphism torsion.

Keywords
zeta function, Reidemeister torsion, $K\mkern-2mu$-theory of endomorphisms
Mathematical Subject Classification
Primary: 19J10, 57Q10
Secondary: 18F25
Milestones
Received: 23 November 2020
Revised: 8 July 2021
Accepted: 18 October 2021
Published: 20 June 2022
Authors
John R. Klein
Department of Mathematics
Wayne State University
Detroit, MI
United States
Cary Malkiewich
Department of Mathematical Sciences
Binghamton University
Binghamton, NY
United States