Counting closed orbits of gradients of circle-valued maps

Details Counting closed orbits of gradients of circle-valued maps Counting closed orbits of gradients of circle-valued maps

2001-04-28

arxiv.org/abs/math/0104273v2

Algebra i Analiz, v.14 (2002), No.3 (in Russian). Engl. Transl: St. Petersburg Math. J. v.14 (2003), No 3.

Mathematics

Differential Geometry

46 pages, Latex file, revised for publication. To appear in Sankt Petersburg Math. Journal

Scientific paper

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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