Refinement of Novikov–Betti numbers and of Novikov homology provided by an angle valued map

To a pair $(X,f)$, $X$ compact ANR and $f\colon X\to \mathbb S^1$ a continuous angle valued map, $\kappa$ a field, and a nonnegative integer $r$, one assigns a finite configuration of complex numbers $z$ with multiplicities $\delta^f_r(z)$ and a finite configuration of free $\kappa[t^{-1}, t]$-modules $\hat \delta^f_r$ of rank $\delta^ f_r(z)$ 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 $\delta^f_r$ refines the Novikov–Betti number in dimension $r$ and the configuration $\hat \delta^f_r$ refines the Novikov homology in dimension $r$ associated with the cohomology class defined by $f$. In the case of the field $\kappa= \mathbb C$, the configuration $\hat \delta^f_r$ provides by “von-Neumann completion” of a configuration $\hat{\hat \delta}^f_r$ of mutually orthogonal closed Hilbert submodules of the $L_2$-homology of the infinite cyclic cover of $X$ determined by the map $f$, which is an $L^\infty(\mathbb S^1)$-Hilbert module.