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Refinement of Novikov–Betti Numbers and of Novikov Homology Provided by an Angle Valued Map

Abstract

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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References

  1. 1.

    D. Burghelea, Linear Relations, Monodromy and Jordan Cells of a Circle Valued Map, arXiv:1501. 02486.

  2. 2.

    D. Burghelea, Refinements of Betti Numbers Provided by a Real Valued Map, arXiv:1501.01012.

  3. 3.

    D. Burghelea, Refinements of Betti Numbers Provided by an Angle Valued Map (in preparation).

  4. 4.

    D. Burghelea, Refinements for Novikov–Betti Numbers = L2-Betti Numbers of (X, ξ ∈ H1(X;ℤ)) Induced by an Angle Valued Map f : X → 𝕊1 (in preparation).

  5. 5.

    D. Burghelea and T. K. Dey, “Persistence for circle-valued maps,” Discrete Comput. Geom., 50, No. 1, 69–98 (2013); arXiv:1104.5646.

  6. 6.

    D. Burghelea and S. Haller, Topology of Angle Valued Maps, Bar Codes and Jordan Blocks, arXiv: 1303.4328, Max Plank preprints.

  7. 7.

    G. Carlsson, V. de Silva, and D. Morozov, “Zigzag persistent homology and real-valued functions,” in: SCG ’09 Proc. of the 25th Annual Symposium on Computational Geometry, ACM, New York (2009), pp. 247–256.

  8. 8.

    T. A. Chapman, Lectures on Hilbert Cube Manifolds, CBMS Reg. Conf. Ser. Math., Vol. 28, Amer. Math. Soc., Providence (1976).

  9. 9.

    D. Cohen-Steiner, H. Edelsbrunner, and J. L. Harer, “Stability of persistence diagrams,” Discrete Comput. Geom., 37, 103–120 (2007).

    MathSciNet  Article  Google Scholar 

  10. 10.

    R. J. Daverman and J. J. Walsh “A ghastly generalized n-manifold,” Illinois J. Math., 25, No. 4, 555–576 (1981).

    MathSciNet  Article  Google Scholar 

  11. 11.

    W. Lück, “Hilbert modules and modules over finite von Neumann algebras and applications to L2 invariants,” Math. Ann., 309, 247–285 (1997).

    MathSciNet  Article  Google Scholar 

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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). https://doi.org/10.1007/s10958-020-04908-9

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