Counting closed orbits of gradients of circle-valued maps

2001/04/30 by A. Pajitnov
Mathematics · #math.DG

paper · pdf

published as Algebra i Analiz, v.14 (2002), No.3 (in Russian). Engl. Transl: St. Petersburg Math. J. v.14 (2003), No 3. · 46 pages, Latex file, revised for publication. To appear in Sankt Petersburg Math. Journal

arxiv created 2002/03/05 · arxiv updated 2009/11/30

Abstract

Let M be a closed connected manifold, f be a Morse map from M to a circle, v be a gradient-like vector field satisfying the transversality condition. The Novikov construction associates to these data a chain complex C_*=C_*(f,v). There is a chain homotopy equivalence between C_* and completed simplicial chain complex of the corresponding infinite cyclic covering of M. The first main result of the paper is the construction of a functorial chain homotopy equivalence between these two complexes. The second main result states that the torsion of this chain homotopy equivalence equals to the Lefschetz zeta function of the gradient flow, if v has only hyperbolic closed orbits.

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