Alternative to Morse-Novikov Theory for closed 1-form (I)

Abstract

This paper extends the Alternative to Morse-Novikov theory we have proposed in Burghelea (New topological invariants for real- and angle valued maps, World Scientific, Hackensack, 2018) from real- and angle-valued map to closed 1-forms. For a topological closed 1-form on a compact ANR (= absolute neighborhood retract), a concept generalizing closed differential 1-form on a compact manifold, under the mild hypothesis of tameness, a field and a non-negative integer we propose two configurations of points, the first on the real line the second on the positive real line, which recover Novikov-Betti numbers and the Novikov complex associated with a Morse closed 1-form with non-degenerated zeros. Precisely, the sum of the multiplicities of the points in the support of the first configuration which correspond to the integer r equals the r-th Novikov-Betti number and that of the points in the support of second configuration which corresponds to the integer r equals the rank of the boundary map in the Novikov complex. We formulate the basic properties of these configurations, the stability property and the Poincare duality property when the compact ANR is a closed orientable topological manifold, which in full generality will be proven in the second and third part of this work.