Geodesic
Refinement of Novikov–Betti numbers and of Novikov homology provided by an angle valued map
Fundamentalʹnaâ i prikladnaâ matematika, Tome 21 (2016) no. 6, pp. 93-113 
D. Burghelea. Refinement of Novikov–Betti numbers and of Novikov homology provided by an angle valued map. Fundamentalʹnaâ i prikladnaâ matematika, Tome 21 (2016) no. 6, pp. 93-113. http://geodesic.mathdoc.fr/item/FPM_2016_21_6_a3/
@article{FPM_2016_21_6_a3,
     author = {D. Burghelea},
     title = {Refinement of {Novikov{\textendash}Betti} numbers and of {Novikov} homology provided by an angle valued map},
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     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2016_21_6_a3/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] Burghelea D., Linear relations, monodromy and Jordan cells of a circle valued map, arXiv: 1501.02486

[2] Burghelea D., Refinements of Betti numbers provided by a real valued map, arXiv: 1501.01012

[3] Burghelea D., Refinements for Novikov–Betti numbers = $L_2$-Betti numbers of $\bigl(X,\xi\in H^1(X;\mathbb Z)\bigr)$ induced by an angle valued map $f\colon X\to \mathbb S^1$, In preparation

[4] Burghelea D., Refinements of Betti numbers provided by an angle valued map, In preparation

[5] Burghelea D., Dey T. K., “Persistence for circle-valued maps”, Discrete Comput. Geom., 50:1 (2013), 69–98, arXiv: 1104.5646 | DOI | MR | Zbl

[6] Burghelea D., Haller S., Topology of angle valued maps, bar codes and Jordan blocks, Max Plank preprints, arXiv: 1303.4328

[7] Carlsson G., de Silva V., Morozov D., “Zigzag persistent homology and real-valued functions”, Proc. of the 25th Annual Symposium on Computational Geometry, SCG '09, ACM, New York, 2009, 247–256 | DOI | Zbl

[8] Chapman T. A., Lectures on Hilbert Cube Manifolds, CBMS Reg. Conf. Ser. Math., 28, Amer. Math. Soc., Providence, 1976 | DOI | MR | Zbl

[9] Cohen-Steiner D., Edelsbrunner H., Harer J. L., “Stability of persistence diagrams”, Discrete Comput. Geom., 37 (2007), 103–120 | DOI | MR | Zbl

[10] Daverman R. J., Walsh J. J., “A ghastly generalized $n$-manifold”, Illinois J. Math., 25:4 (1981), 555–576 | MR | Zbl

[11] Lück W., “Hilbert modules and modules over finite von Neumann algebras and applications to $L^2$ invariants”, Math. Ann., 309 (1997), 247–285 | DOI | MR | Zbl