Zeros of closed 1-forms, homoclinic orbits, and Lusternik - Schnirelman theory

In this paper we study topological lower bounds on the number of zeros of closed 1-forms without Morse type assumptions. We prove that one may always find a representing closed 1-form having at most one zero. We introduce and study a generalization $cat(X,\xi)$ of the notion of Lusternik - Schnirelman category, depending on a topological space $X$ and a cohomology class $\xi\in H^1(X;\R)$. We prove that any closed 1-form has at least $cat(X,\xi)$ zeros assuming that it admits a gradient-like vector field with no homoclinic cycles. We show that the number $cat(X,\xi)$ can be estimated from below in terms of the cup-products and higher Massey products. This paper corrects some statements made in my previous papers on this subject.Comment: 34 pages. A refernce adde

Combinatorial invariants computing the Ray-Singer analytic torsion

It is shown that for any piecewise-linear closed orientable manifold of odd dimension there exists an invariantly defined metric on the determinant line of cohomology with coefficients in an arbitrary flat bundle E over the manifold (E is not required to be unimodular). The construction of this metric (called Poincare - Reidemeister metric) is purely combinatorial; it combines the standard Reidemeister type construction with Poincare duality. The main result of the paper states that the Poincare-Reidemeister metric computes combinatorially the Ray-Singer metric. It is shown also that the Ray-Singer metrics on some relative determinant lines can be computed combinatorially (including the even-dimensional case) in terms of metrics determined by correspondences.Comment: Amstex, 19 pages, to appear in "Differential Geometry and Applications

Absolute torsion and eta-invariant

In a recent joint work with V. Turaev (cf. math.DG/9810114) we defined a new concept of combinatorial torsion which we called absolute torsion. Compared with the classical Reidemeister torsion it has the advantage of having a well-defined sign. Also, the absolute torsion is defined for arbitrary orientable flat vector bundles, and not only for unimodular ones, as is classical Reidemeister torsion. In this paper I show that the sign behavior of the absolute torsion, under a continuous deformation of the flat bundle, is determined by the eta-invariant and the Pontrjagin classes.Comment: 10 page

Lusternik - Schnirelman theory and dynamics

In this paper we study a new topological invariant \Cat(X,\xi), where $X$ is a finite polyhedron and $\xi\in H^1(X;\R)$ is a real cohomology class. \Cat(X,\xi) is defined using open covers of $X$ with certain geometric properties; it is a generalization of the classical Lusternik -- Schnirelman category. We show that \Cat(X,\xi) depends only on the homotopy type of $(X,\xi)$. We prove that \Cat(X,\xi) allows to establish a relation between the number of equilibrium states of dynamical systems and their global dynamical properties (such as existence of homoclinic cycles and the structure of the set of chain recurrent points). In the paper we give a cohomological lower bound for \Cat(X,\xi), which uses cup-products of cohomology classes of flat line bundles with monodromy described by complex numbers, which are not Dirichlet units.Comment: 20 page