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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[t1, t]-modules \( {\hat{\delta}}_r^f \) of rank \( {\delta}_r^f \) (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 \( {\delta}_r^f \) refines the Novikov–Betti number in dimension r, and the configuration \( {\hat{\delta}}_r^f \) 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 \( {\hat{\delta}}_r^f \) provides by “von-Neumann completion” of a configuration \( {\hat{\hat{\delta}}}_r^f \) 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.

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Correspondence to D. Burghelea.

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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 21, No. 6, pp. 93–113, 2016.

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Burghelea, D. Refinement of Novikov–Betti Numbers and of Novikov Homology Provided by an Angle Valued Map. J Math Sci 248, 728–742 (2020).

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