# Novikov homology, jump loci and Massey products

Toshitake Kohno; Andrei Pajitnov

Open Mathematics (2014)

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

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topToshitake 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 -

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