TY - JOUR

T1 - Morse-Novikov critical point theory, Cohn localization and Dirichlet units

AU - Farber, M.

PY - 1999/11

Y1 - 1999/11

N2 - 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 minimum number of zeros 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.

AB - 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 minimum number of zeros 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.

KW - Closed 1-forms

KW - Cohn localization

KW - Morse theory

KW - Novikov inequalities

UR - http://www.scopus.com/inward/record.url?scp=22844457130&partnerID=8YFLogxK

U2 - 10.1142/S0219199799000171

DO - 10.1142/S0219199799000171

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AN - SCOPUS:22844457130

SN - 0219-1997

VL - 1

SP - 467

EP - 495

JO - Communications in Contemporary Mathematics

JF - Communications in Contemporary Mathematics

IS - 4

ER -