Lefschetz zeta functions of gradients of circle-valued maps
by
Andrei Pajitnov
Universite de Nantes, France

Let f be a Morse map from a closed manifold M to a circle and let v be a gradient of f. The subject of the talk is the dynamics of the flow generated by v in connection with the topology of the manifold.

We assume that the closed orbits of v are hyperbolic. There is a naturally arising Lefschetz zeta function, which counts closed orbits of v (each orbit is counted with a weight depending on its index and homology class). This zeta function Z(v) is a power series in one variable. We prove that Z(v) is a rational function if v is C⁰-generic.

There is another geometric object associated to v via counting its orbits, namely Novikov complex. Let L be the ring of Laurent power series in one variable with finite negative part. The Novikov complex C_*(v) is a chain complex of free modules over L, generated by the critical points of f  (v is supposed to satisfy the transversality assumption). The boundary operators are defined via counting the orbits of v joining the critical points of f  (similarly to the case of the classical Morse complex of a real-valued function). The complex C_*(v) is homotopy equivalent to the completed simplicial chain complex of the infinite cyclic covering of M corresponding to f. We prove that for a C⁰-generic v the Lefschetz zeta function of v equals to the torsion of this homotopy equivalence.

Date received: May 29, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cacy-19.