$K$-theoretic torsion and the zeta function

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 non-trivial endomorphism torsion. Contents1. Introduction 2. Preliminaries 3. Higher endomorphism torsion 4. The higher zeta function 5. A generalization of Milnor's identity 6. Chain complexes 7. The non-linear setting 8. Families of endomorphisms Appendix A. Fundamental theorems of endomorphism K-theory Appendix B. Endomorphism torsion and boundary map Appendix C. The sphere theorem for End A and End S A