  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.
