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

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,\k), and the \kπ_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,\k), thereby generalizing a result of Cohen and Orlik.

38 pages, 1 figure (section 10 of version 1 has been significantly expanded into a separate paper, available at arXiv:0901.0105); accepted for publication in the Transactions of the American Mathematical Society

References in corpus (5)

• Toric complexes and Artin kernels
• On the monodromy action on Milnor fibers of graphic arrangements
• Chamber basis of the Orlik-Solomon algebra and Aomoto complex
• Algebraic monodromy and obstructions to formality
• Alexander polynomials: Essential variables and multiplicities

Cited by in corpus (26)

• The Milnor fibration of a hyperplane arrangement: from modular resonance to algebraic monodromy
• Toric complexes and Artin kernels
• On the monodromy action on Milnor fibers of graphic arrangements
• Geometric and algebraic aspects of 1-formality
• Chamber basis of the Orlik-Solomon algebra and Aomoto complex
• Algebraic monodromy and obstructions to formality
• Resonance varieties and Dwyer-Fried invariants
• Abelian duality and propagation of resonance
• Double coverings of arrangement complements and $2$-torsion in Milnor fiber homology
• Cup products, lower central series, and holonomy Lie algebras
• Tait colorings, and an instanton homology for webs and foams
• Chen ranks and resonance varieties of the upper McCool groups
• Homology graph of real arrangements and monodromy of Milnor Fiber
• Combinatorial stratifications and minimality of 2-arrangements
• On the monodromy of Milnor fibers of hyperplane arrangements
• Szczarba's twisting cochain is comultiplicative
• Multiplicativity and nonrealizable equivariant chain complexes
• A Levine-Tristram invariant for knotted tori
• Cohomology, Bocksteins, and resonance varieties in characteristic 2
• Milnor fibrations of arrangements with trivial algebraic monodromy
• On the topology of the Milnor fibration of a hyperplane arrangement
• On the second nilpotent quotient of higher homotopy groups, for hypersolvable arrangements
• $L^2$-type invariants and cohomology jump loci for complex smooth quasi-projective varieties
• Vanishing results for the Aomoto complex of real hyperplane arrangements via minimality
• Resonant bands, Aomoto complex, and real 4-nets
• A survey of combinatorial aspects in the topology of complex hyperplane arrangements