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 Fundam. Prikl. Mat., 2016, Volume 21, Issue 6, Pages 93–113 (Mi fpm1770)

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

D. Burghelea

The Ohio State University, Columbus, Ohio

Abstract: 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.

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UDC: 515.142

Citation: D. Burghelea, “Refinement of Novikov–Betti numbers and of Novikov homology provided by an angle valued map”, Fundam. Prikl. Mat., 21:6 (2016), 93–113

Citation in format AMSBIB
\Bibitem{Bur16} \by D.~Burghelea \paper Refinement of Novikov--Betti numbers and of Novikov homology provided by an angle valued map \jour Fundam. Prikl. Mat. \yr 2016 \vol 21 \issue 6 \pages 93--113 \mathnet{http://mi.mathnet.ru/fpm1770}