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Transactions of the American Mathematical Society

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

Authors: Stefan Papadima and Alexander I. Suciu
Journal: Trans. Amer. Math. Soc. 362 (2010), 2685-2721
MSC (2010): Primary 55N25, 55T99; Secondary 20J05, 57M05
Published electronically: December 15, 2009
MathSciNet review: 2584616
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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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Additional Information

Stefan Papadima
Affiliation: Institute of Mathematics Simion Stoilow, P.O. Box 1-764, RO-014700 Bucharest, Romania

Alexander I. Suciu
Affiliation: Department of Mathematics, Northeastern University, Boston, Massachusetts 02115

Keywords: Equivariant chain complex, $I$-adic filtration, spectral sequence, twisted homology, minimal cell complex, Aomoto complex, Betti numbers
Received by editor(s): September 29, 2008
Published electronically: December 15, 2009
Additional Notes: The first author was partially supported by the CEEX Programme of the Romanian Ministry of Education and Research, contract 2-CEx 06-11-20/2006
The second author was partially supported by NSF grant DMS-0311142
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.