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For a continuous angle-valued map defined on a compact ANR, a fixed field andany degree one proposes a refinement of the Novikov-Betti number and of theNovikov homology of the pair consisting of the ANR and the degree one integralcohomology class represented by the map. For each degree the first refinementis a configuration of points with multiplicity located in the punctured complexplane whose total cardinality is the Novikov-Betti number of the pair. Thesecond refinement is a configuration of submodules of the Novikov homologywhose direct sum is isomorphic to the Novikov homology and which has the samesupport as the first configuration. When the field is a the field of complexnumbers the second configuration is convertible into a configuration ofmutually orthogonal closed Hilbert submodules of the L2-homology of theinfinite cyclic cover of the ANR defined by the angle-valued map. One discussesthe properties of these configurations namely, robustness with respect tocontinuous perturbation of the angle-valued map and the Poincar\'e Duality andone derives some computational applications in topology. The main resultsparallel the results for the case of real-valued map but with Novikov homologyand Novikov-Betti numbers replacing standard homology and standard Bettinumbers.

A refinement of Betti numbers and homology in the presence of a continuous function II (the case of an angle valued map)