A Functional Equation for the Lefschetz Zeta Functions of Infinite Cyclic Coverings with an Application to Knot Theory

The Weil conjecture is a delightful theorem for algebraic varieties on finite fields and an important model for dynamical zeta functions. In this paper, we prove a functional equation of Lefschetz zeta functions for infinite cyclic coverings which is analogous to the Weil conjecture. Applying this functional equation to knot theory, we obtain a new view point on the reciprocity of the Alexander polynomial of a knot. INTRODUCTION The Lefschetz zeta function is one of the dynamical zeta functions. Dynamical zeta functions are developed in order to study the number of fixed points or periodic points. These zeta functions are motivated by the Weil conjecture. Therefore, studying dynamical zeta functions is usually modeled after Weil conjecture. On the other hand, dynamical zeta functions are useful in geometric topology. For example, if you fix a map to a geometrical one, it can be related to topological invariants (e.g. [5, 8, 9, 16]). That is, a property of a topological invariant can be regarded as a property of the dynamical zeta function. Grothendieck (after the works of Weil and Serre) observed that the Weil conjecture should be the consequence of a certain good cohomology theory, which is called the Weil cohomology, and in particular the functional equation should be derived from the Poincaré duality of the cohomology (see section 1.2). In this paper, we prove an analogous functional equation of Lefschetz zeta functions by following his idea, and apply it to knot theory. To do that, we have to resolve the following two problems: (1) the infinite cyclic covering of a knot compliment is a non-compact odddimensional manifold but the Weil cohomology requires the even-dimensional Poincaré duality, and (2) this manifold has the boundary, so we need to deal with the relative version of Poincaré duality (Lefschetz-Poincaré duality). The answer of the first is Milnor’s duality theorem for infinite cyclic coverings [19]. The device for the second is the Lefschetz zeta function for the boundary, which is defined in Definition 9. With those tools, the following theorem is proved in Section 2. 2000 Mathematics Subject Classification. Primary 57M27; Secondary 37C30. The author was supported in part by JSPS fellowship for young scientists. 1