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2014 | 12 | 9 | 1285-1304
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Novikov homology, jump loci and Massey products

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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).
  • University of Tokyo
  • Université de Nantes, Faculté des Sciences
  • [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
  • [2] Benson C., Gordon C.S., Kähler structures on compact solvmanifolds, Proc. Amer. Math. Soc., 1990, 108(4), 971–980
  • [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
  • [4] Deligne P., Griffiths Ph., Morgan J., Sullivan D., Real homotopy theory of Kähler manifolds, Invent. Math., 1975, 29(3), 245–274
  • [5] Dimca A., Papadima S., Nonabelian cohomology jump loci from an analytic viewpoint, preprint avaliable at
  • [6] Farber M., Lusternik-Schnirelman theory for closed 1-forms, Comment. Math. Helv., 2000, 75(1), 156–170
  • [7] Farber M., Topology of closed 1-forms and their critical points, Topology, 2001, 40(2), 235–258
  • [8] Friedl S., Vidussi S., A vanishing theorem for twisted Alexander polynomials with applications to symplectic 4-manifolds, preprint available at
  • [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
  • [10] Hu S., Homotopy Theory, Pure Appl. Math., 8, Academic Press, New York-London, 1959
  • [11] Kasuya H., Minimal models, formality, and hard Lefschetz properties of solvmanifolds with local systems, J. Differential Geom., 2013, 93(2), 269–297
  • [12] Massey W.S., Exact couples in algebraic topology, I, II, Ann. Math., 1952, 56(2), 363–396
  • [13] Mostow G.D., Cohomology of topological groups and solvmanifolds, Ann. Math., 1961, 73(1), 20–48
  • [14] Novikov S.P., Multivalued functions and functionals. An analogue of the Morse theory, Soviet Math. Dokl., 1981, 24, 222–226
  • [15] Novikov S.P., Bloch homology. Critical points of functions and closed 1-forms, Soviet Math. Dokl., 1986, 33(2), 551–555
  • [16] Pajitnov A.V., Novikov homology, twisted Alexander polynomials, and Thurston cones, St. Petersburg Math. J., 2007, 18(5), 809–835
  • [17] Papadima S., Suciu A.I., Bieri-Neumann-Strebel-Renz invariants and homology jumping loci, Proc. Lond. Math. Soc., 2010, 100(3), 795–834
  • [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
  • [19] Pazhitnov A.V., An analytic proof of the real part of Novikov’s inequalities, Soviet Math. Dokl., 1987, 35(2), 456–457
  • [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
  • [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
  • [22] Sawai H., A construction of lattices on certain solvable Lie groups, Topology Appl., 2007, 154(18), 3125–3134
  • [23] Simpson C.T., Higgs bundles and local systems, Inst. Hautes Études Sci. Publ. Math., 1992, 75, 5–95
  • [24] Sullivan D., Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math., 1977, 47, 269–331
  • [25] Wells R.O. Jr., Differential Analysis on Complex Manifolds, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Englewood Cliffs, 1973
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