The spectral sequence of an equivariant chain complex and homology with local coefficients

Papadima, Stefan; Suciu, Alexander

doi:10.1090/S0002-9947-09-05041-7

The spectral sequence of an equivariant chain complex and homology with local coefficients
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by Stefan Papadima and Alexander I. Suciu PDF

Trans. Amer. Math. Soc. 362 (2010), 2685-2721 Request permission

Abstract:

We study the spectral sequence associated to the filtration by powers of the augmentation ideal on the (twisted) equivariant chain complex of the universal cover of a connected CW-complex $X$. In the process, we identify the $d^1$ differential in terms of the coalgebra structure of $H_*(X,\Bbbk )$ and the $\Bbbk \pi _1(X)$-module structure on the twisting coefficients. In particular, this recovers in dual form a result of Reznikov on the mod $p$ cohomology of cyclic $p$-covers of aspherical complexes. This approach provides information on the homology of all Galois covers of $X$. It also yields computable upper bounds on the ranks of the cohomology groups of $X$, with coefficients in a prime-power order, rank one local system. When $X$ admits a minimal cell decomposition, we relate the linearization of the equivariant cochain complex of the universal abelian cover to the Aomoto complex, arising from the cup-product structure of $H^*(X,\Bbbk )$, thereby generalizing a result of Cohen and Orlik.

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