Kavli Affiliate: Toshitake Kohno
| First 5 Authors: Toshitake Kohno, Andrei Pajitnov, , ,
| Summary:
Let X be a finite CW-complex, denote its fundamental group by G. Let R be an
n-dimensional complex repesentation of G. Any element A of the first cohomology
group of X with complex coefficients gives rise to the exponential deformation
of the representation R, which can be considered as a curve in the space of
representations. We show that the cohomology of X with local coefficients
corresponding to the generic point of this curve is computable from a spectral
sequence starting from the cohomology of X with R-twisted coefficients. We
compute the differentials of the spectral sequence in terms of Massey products.
We show that the spectral sequence degenerates in case when X is a Kaehler
manifold, and the representation R is semi-simple.
If A is a real cohomology class, one associates to the triple (X,R,A) 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 above local system. We investigate the dependance of these numbers on A and
prove that they are constant in the complement to a finite number of integral
hyperplanes in the first cohomology group.
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