Refinement of Novikov–Betti Numbers and of Novikov Homology Provided by an Angle Valued Map

To a pair (X, f), where X is a compact ANR and f : X → S1 is a continuous angle valued map, a field κ, and a nonnegative integer r, one assigns a finite configuration of complex numbers z with multiplicities δfr(z) and a finite configuration of free κ[t−1, t]-modules δ̂rf of rank δfr (z) indexed by the same numbers z. This is in analogy with the configuration of eigenvalues and of generalized eigenspaces of a linear operator in a finite-dimensional complex vector space. The configuration δrf refines the Novikov–Betti number in dimension r, and the configuration δ̂rf refines the Novikov homology in dimension r associated with the cohomology class defined by f. In the case of the field κ = C, the configuration δ̂rf provides by “von-Neumann completion” of a configuration δ̂fr of mutually orthogonal closed Hilbert submodules of the L2-homology of the infinite cyclic cover of X determined by the map f, which is an L∞(S1)-Hilbert module. © 2020, Springer Science+Business Media, LLC, part of Springer Nature.