For a continuous angle-valued map defined on a compact ANR , a field and any integer , one proposes a refinement of the Novikov–Betti numbers of the pair and a refinement of the Novikov homology of , where denotes the integral degree one cohomology class represented by . The refinement is a configuration of points, with multiplicity located in identified to , whose total cardinality is the Novikov–Betti number of the pair. The refinement is a configuration of submodules of the Novikov homology whose direct sum is isomorphic to the Novikov homology and with the same support as of . When , the configuration is convertible into a configuration of mutually orthogonal closed Hilbert submodules of the –homology of the infinite cyclic cover of defined by , which is an –Hilbert module. One discusses the properties of these configurations, namely robustness with respect to continuous perturbation of the angle-values map and the Poincaré duality and one derives some computational applications in topology. The main results parallel the results for the case of real-valued map but with Novikov homology and Novikov–Betti numbers replacing standard homology and standard Betti numbers.
"A refinement of Betti numbers and homology in the presence of a continuous function, II: The case of an angle-valued map." Algebr. Geom. Topol. 18 (5) 3037 - 3087, 2018. https://doi.org/10.2140/agt.2018.18.3037