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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo


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Volume 12, Issue 9

Issues

Volume 13 (2015)

Novikov homology, jump loci and Massey products

Toshitake Kohno
  • Kavli IPMU (WPI), Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan
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/ Andrei Pajitnov
  • Laboratoire Mathématiques Jean Leray UMR 6629, Université de Nantes, Faculté des Sciences, 2, rue de la Houssinière, 44072, Nantes, Cedex, France
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Published Online: 2014-05-08 | DOI: https://doi.org/10.2478/s11533-014-0413-2

Abstract

Let X be a finite CW complex, and ρ: π 1(X) → GL(l, ℂ) a representation. Any cohomology class α ∈ H 1(X, ℂ) gives rise to a deformation γ t of ρ defined by γ t (g) = ρ(g) exp(t〈α, g〉). We show that the cohomology of X with local coefficients γ gen corresponding to the generic point of the curve γ is computable from a spectral sequence starting from H*(X, ρ). We compute the differentials of the spectral sequence in terms of the Massey products and show that the spectral sequence degenerates in case when X is a Kähler manifold and ρ is semi-simple.

If α ∈ H 1(X, ℝ) one associates to the triple (X, ρ, α) the twisted Novikov homology (a module over the Novikov ring). We show that the twisted Novikov Betti numbers equal the Betti numbers of X with coefficients in the local system γ gen. We investigate the dependence of these numbers on α and prove that they are constant in the complement to a finite number of proper vector subspaces in H 1(X, ℝ).

MSC: 14F40; 53B35; 55N25; 58E05

Keywords: Novikov homology; Cohomology with twisted coefficients; Spectral sequence; Massey products; Kähler manifolds

  • [1] Arapura D., Higgs line bundles, Green-Lazarsfeld sets, and maps of Kähler manifolds to curves, Bull. Amer. Math. Soc. (N.S.), 1992, 26(2), 310–314 http://dx.doi.org/10.1090/S0273-0979-1992-00283-5CrossrefGoogle Scholar

  • [2] Benson C., Gordon C.S., Kähler structures on compact solvmanifolds, Proc. Amer. Math. Soc., 1990, 108(4), 971–980 Google Scholar

  • [3] Bousfield A.K., Guggenheim V.K.A.M., On PL de Rham Theory and Rational Homotopy Type, Mem. Amer. Math. Soc., 179, American Mathematical Society, Providence, 1976 CrossrefGoogle Scholar

  • [4] Deligne P., Griffiths Ph., Morgan J., Sullivan D., Real homotopy theory of Kähler manifolds, Invent. Math., 1975, 29(3), 245–274 http://dx.doi.org/10.1007/BF01389853CrossrefGoogle Scholar

  • [5] Dimca A., Papadima S., Nonabelian cohomology jump loci from an analytic viewpoint, preprint avaliable at http://arxiv.org/abs/1206.3773 Google Scholar

  • [6] Farber M., Lusternik-Schnirelman theory for closed 1-forms, Comment. Math. Helv., 2000, 75(1), 156–170 http://dx.doi.org/10.1007/s000140050117Google Scholar

  • [7] Farber M., Topology of closed 1-forms and their critical points, Topology, 2001, 40(2), 235–258 http://dx.doi.org/10.1016/S0040-9383(99)00059-2CrossrefGoogle Scholar

  • [8] Friedl S., Vidussi S., A vanishing theorem for twisted Alexander polynomials with applications to symplectic 4-manifolds, preprint available at http://arxiv.org/abs/1205.2434 Google Scholar

  • [9] Goda H., Pajitnov A.V., Twisted Novikov homology and circle-valued Morse theory for knots and links, Osaka J. Math., 2005, 42(3), 557–572 Google Scholar

  • [10] Hu S., Homotopy Theory, Pure Appl. Math., 8, Academic Press, New York-London, 1959 Google Scholar

  • [11] Kasuya H., Minimal models, formality, and hard Lefschetz properties of solvmanifolds with local systems, J. Differential Geom., 2013, 93(2), 269–297 Google Scholar

  • [12] Massey W.S., Exact couples in algebraic topology, I, II, Ann. Math., 1952, 56(2), 363–396 http://dx.doi.org/10.2307/1969805CrossrefGoogle Scholar

  • [13] Mostow G.D., Cohomology of topological groups and solvmanifolds, Ann. Math., 1961, 73(1), 20–48 http://dx.doi.org/10.2307/1970281CrossrefGoogle Scholar

  • [14] Novikov S.P., Multivalued functions and functionals. An analogue of the Morse theory, Soviet Math. Dokl., 1981, 24, 222–226 Google Scholar

  • [15] Novikov S.P., Bloch homology. Critical points of functions and closed 1-forms, Soviet Math. Dokl., 1986, 33(2), 551–555 Google Scholar

  • [16] Pajitnov A.V., Novikov homology, twisted Alexander polynomials, and Thurston cones, St. Petersburg Math. J., 2007, 18(5), 809–835 http://dx.doi.org/10.1090/S1061-0022-07-00975-2CrossrefGoogle Scholar

  • [17] Papadima S., Suciu A.I., Bieri-Neumann-Strebel-Renz invariants and homology jumping loci, Proc. Lond. Math. Soc., 2010, 100(3), 795–834 http://dx.doi.org/10.1112/plms/pdp045CrossrefGoogle Scholar

  • [18] Papadima S., Suciu A.I., The spectral sequence of an equivariant chain complex and homology with local coefficients, Trans. Amer. Math. Soc., 2010, 362(5), 2685–2721 http://dx.doi.org/10.1090/S0002-9947-09-05041-7Web of ScienceCrossrefGoogle Scholar

  • [19] Pazhitnov A.V., An analytic proof of the real part of Novikov’s inequalities, Soviet Math. Dokl., 1987, 35(2), 456–457 Google Scholar

  • [20] Pazhitnov A.V., Proof of Novikov’s conjecture on homology with local coefficients over a field of finite characteristic, Soviet Math. Dokl., 1988, 37(3), 824–828 Google Scholar

  • [21] Pazhitnov A.V., On the sharpness of inequalities of Novikov type for manifolds with a free abelian fundamental group, Math. USSR-Sb., 1990, 68(2), 351–389 http://dx.doi.org/10.1070/SM1991v068n02ABEH001933CrossrefGoogle Scholar

  • [22] Sawai H., A construction of lattices on certain solvable Lie groups, Topology Appl., 2007, 154(18), 3125–3134 http://dx.doi.org/10.1016/j.topol.2007.08.006CrossrefWeb of ScienceGoogle Scholar

  • [23] Simpson C.T., Higgs bundles and local systems, Inst. Hautes Études Sci. Publ. Math., 1992, 75, 5–95 http://dx.doi.org/10.1007/BF02699491CrossrefGoogle Scholar

  • [24] Sullivan D., Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math., 1977, 47, 269–331 http://dx.doi.org/10.1007/BF02684341CrossrefGoogle Scholar

  • [25] Wells R.O. Jr., Differential Analysis on Complex Manifolds, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Englewood Cliffs, 1973 Google Scholar

About the article

Published Online: 2014-05-08

Published in Print: 2014-09-01


Citation Information: Open Mathematics, Volume 12, Issue 9, Pages 1285–1304, ISSN (Online) 2391-5455, DOI: https://doi.org/10.2478/s11533-014-0413-2.

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