Morse-Novikov critical point theory, Cohn localization and Dirichlet units

Abstract

In this paper we construct a Universal chain complex, counting zeros of closed 1-forms on a manifold. The Universal complex is a refinement of the well known Novikov complex; it relates the homotopy type of the manifold, after a suitable noncommutative localization, with the numbers of zeros of different indices which may have closed 1-forms within a given cohomology class. The Main Theorem of the paper generalizes the result of a joint paper with A. Ranicki, which treats the special case of closed 1-forms having integral cohomology classes. The present paper also describes a number of new inequalities, giving topological lower bounds on the number of zeroes of closed 1-forms. In particular, such estimates are provided by the homology of flat line bundles with monodromy described by complex numbers which are not Dirichlet units.