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