## 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.## References

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## Additional Information

**Stefan Papadima**- Affiliation: Institute of Mathematics Simion Stoilow, P.O. Box 1-764, RO-014700 Bucharest, Romania
- Email: Stefan.Papadima@imar.ro
**Alexander I. Suciu**- Affiliation: Department of Mathematics, Northeastern University, Boston, Massachusetts 02115
- MR Author ID: 168600
- ORCID: 0000-0002-5060-7754
- Email: a.suciu@neu.edu
- 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 - © Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**362**(2010), 2685-2721 - MSC (2010): Primary 55N25, 55T99; Secondary 20J05, 57M05
- DOI: https://doi.org/10.1090/S0002-9947-09-05041-7
- MathSciNet review: 2584616