A refinement of Betti numbers and homology in the presence of a continuous function, II: The case of an angle-valued map (Q1673648)

In a previous paper [Algebr. Geom. Topol. 17, No. 4, 2051--2080 (2017; Zbl 1378.55003)] the author described how, given a continuous real-valued function on a compact ANR \(X\), one can get a refinement of the Betti numbers and homology of \(X\). The refinement creates a configuration of points in complex space, whose total cardinalities are the Betti numbers. The refinement of the homology consists of a set of vector spaces indexed by points in the complex plane with the same support as that of the point configuration. The direct sum of these vector spaces is the homology of \(X\). In this paper the author starts with an ``angle-valued'' map which is simply a map into the circle \(f: X \;\rightarrow \;S^{1}\). By considering the lift \(\tilde{X} \;\rightarrow {\mathbb R}\) and applying similar, but more complex, reasoning the author uses the deck transformations to obtain a \(\kappa [t,t^{-1}]\) structure (where \(\kappa\) is a field) on the homology and obtains results parallel to those of the first paper. As in the first paper a Poincaré duality result is shown for the case where \(X\) is a closed smooth manifold.