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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


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


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

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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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