Cup products, lower central series, and holonomy Lie algebras

Abstract

We generalize basic results relating the associated graded Lie algebra and the holonomy Lie algebra from finitely presented, commutator-relators groups to arbitrary finitely presented groups. In the process, we give an explicit formula for the cup-product in the cohomology of a finite 2-complex, and an algorithm for computing the corresponding holonomy Lie algebra, using a Magnus expansion method. We illustrate our approach with examples drawn from a variety of group-theoretic and topological contexts, such as link groups, one-relator groups, and fundamental groups of orientable Seifert fibered manifolds.

arXiv comment27 pages; accepted for publication in the Journal of Pure and Applied Algebra. arXiv admin note: substantial text overlap with arXiv:1504.08294

References in corpus (12)

• Topology and geometry of cohomology jump loci
• Chen Lie algebras
• The spectral sequence of an equivariant chain complex and homology with local coefficients
• Geometric and algebraic aspects of 1-formality
• Formality properties of finitely generated groups and Lie algebras
• Vanishing resonance and representations of Lie algebras
• Homotopy types of complements of 2-arrangements in R^4
• The pure braid groups and their relatives
• Jump loci in the equivariant spectral sequence
• Pure virtual braids, resonance, and formality
• The Pure Virtual Braid Group Is Quadratic
• Chen ranks and resonance varieties of the upper McCool groups

Cited by in corpus (7)

• Formality properties of finitely generated groups and Lie algebras
• Chen ranks and resonance varieties of the upper McCool groups
• Homology, lower central series, and hyperplane arrangements
• Formality and finiteness in rational homotopy theory
• Taylor expansions of groups and filtered-formality
• Alexander invariants and cohomology jump loci in group extensions
• Strongly isospectral hyperbolic 3-manifolds with nonisomorphic rational cohomology rings