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In this paper we construct a Universal chain complex, counting zeros ofclosed 1-forms on a manifold. The Universal complex is a refinement of the wellknown Novikov complex; it relates the homotopy type of the manifold, after asuitable noncommutative localization, with the numbers of zeros of differentindices which may have closed 1-forms within a given cohomology class. The MainTheorem of the paper generalizes the result of a joint paper with A. Ranicki,which treats the special case of closed 1-forms having integral cohomologyclasses. The present paper also describes a number of new inequalities, givingtopological lower bounds on the number of zeroes of closed 1-forms. Inparticular, such estimates are provided by the homology of flat line bundleswith monodromy described by complex numbers which are not Dirichlet units.

MORSE–NOVIKOV CRITICAL POINT THEORY, COHN LOCALIZATION AND DIRICHLET UNITS