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

Abstract

For $f:X\to {\mathbb{S}}^1$ a continuous angle-valued map defined on a compact ANR $X$, $\kappa$ a field and any integer $r\ge 0$, one proposes a refinement ${\delta }_r^f$ of the Novikov–Betti numbers of the pair $\left(X,{\xi }_f\right)$ and a refinement ${\widehat{\delta }}_r^f$ of the Novikov homology of $\left(X,{\xi }_f\right)$, where ${\xi }_f$ denotes the integral degree one cohomology class represented by $f$. The refinement ${\delta }_r^f$ is a configuration of points, with multiplicity located in ${ℝ}^2∕\mathbb{Z}$ identified to $ℂ\setminus 0$, whose total cardinality is the $r^{\text{ th}}$ Novikov–Betti number of the pair. The refinement ${\widehat{\delta }}_r^f$ is a configuration of submodules of the $r^{\text{ th}}$ Novikov homology whose direct sum is isomorphic to the Novikov homology and with the same support as of ${\delta }_r^f$. When $\kappa =ℂ$, the configuration ${\widehat{\delta }}_r^f$ is convertible into a configuration of mutually orthogonal closed Hilbert submodules of the $L_2$–homology of the infinite cyclic cover of $X$ defined by $f$, which is an $L^{\infty }\left({\mathbb{S}}^1\right)$–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.