Novikov homology, jump loci and Massey products

Toshitake Kohno; Andrei Pajitnov

Open Mathematics (2014)

  • Volume: 12, Issue: 9, page 1285-1304
  • ISSN: 2391-5455

Abstract

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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).

How to cite

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Toshitake Kohno, and Andrei Pajitnov. "Novikov homology, jump loci and Massey products." Open Mathematics 12.9 (2014): 1285-1304. <http://eudml.org/doc/269518>.

@article{ToshitakeKohno2014,
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).},
author = {Toshitake Kohno, Andrei Pajitnov},
journal = {Open Mathematics},
keywords = {Novikov homology; Cohomology with twisted coefficients; Spectral sequence; Massey products; Kähler manifolds; cohomology with twisted coefficients; spectral sequence},
language = {eng},
number = {9},
pages = {1285-1304},
title = {Novikov homology, jump loci and Massey products},
url = {http://eudml.org/doc/269518},
volume = {12},
year = {2014},
}

TY - JOUR
AU - Toshitake Kohno
AU - Andrei Pajitnov
TI - Novikov homology, jump loci and Massey products
JO - Open Mathematics
PY - 2014
VL - 12
IS - 9
SP - 1285
EP - 1304
AB - 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).
LA - eng
KW - Novikov homology; Cohomology with twisted coefficients; Spectral sequence; Massey products; Kähler manifolds; cohomology with twisted coefficients; spectral sequence
UR - http://eudml.org/doc/269518
ER -

References

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  1. [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/S0273-0979-1992-00283-5 Zbl0759.14016
  2. [2] Benson C., Gordon C.S., Kähler structures on compact solvmanifolds, Proc. Amer. Math. Soc., 1990, 108(4), 971–980 Zbl0689.53036
  3. [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 Zbl0338.55008
  4. [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/BF01389853 Zbl0312.55011
  5. [5] Dimca A., Papadima S., Nonabelian cohomology jump loci from an analytic viewpoint, preprint avaliable at http://arxiv.org/abs/1206.3773 
  6. [6] Farber M., Lusternik-Schnirelman theory for closed 1-forms, Comment. Math. Helv., 2000, 75(1), 156–170 http://dx.doi.org/10.1007/s000140050117 Zbl0956.58006
  7. [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)00059-2 Zbl0977.58011
  8. [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 Zbl1285.57007
  9. [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 Zbl1083.57017
  10. [10] Hu S., Homotopy Theory, Pure Appl. Math., 8, Academic Press, New York-London, 1959 
  11. [11] Kasuya H., Minimal models, formality, and hard Lefschetz properties of solvmanifolds with local systems, J. Differential Geom., 2013, 93(2), 269–297 Zbl06201318
  12. [12] Massey W.S., Exact couples in algebraic topology, I, II, Ann. Math., 1952, 56(2), 363–396 http://dx.doi.org/10.2307/1969805 Zbl0049.24002
  13. [13] Mostow G.D., Cohomology of topological groups and solvmanifolds, Ann. Math., 1961, 73(1), 20–48 http://dx.doi.org/10.2307/1970281 Zbl0103.26501
  14. [14] Novikov S.P., Multivalued functions and functionals. An analogue of the Morse theory, Soviet Math. Dokl., 1981, 24, 222–226 Zbl0505.58011
  15. [15] Novikov S.P., Bloch homology. Critical points of functions and closed 1-forms, Soviet Math. Dokl., 1986, 33(2), 551–555 
  16. [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/S1061-0022-07-00975-2 
  17. [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/pdp045 Zbl1273.55003
  18. [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/S0002-9947-09-05041-7 Zbl1195.55005
  19. [19] Pazhitnov A.V., An analytic proof of the real part of Novikov’s inequalities, Soviet Math. Dokl., 1987, 35(2), 456–457 Zbl0647.57025
  20. [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 Zbl0667.55003
  21. [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/SM1991v068n02ABEH001933 Zbl0708.57013
  22. [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.2007.08.006 Zbl1138.53065
  23. [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/BF02699491 Zbl0814.32003
  24. [24] Sullivan D., Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math., 1977, 47, 269–331 http://dx.doi.org/10.1007/BF02684341 Zbl0374.57002
  25. [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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