Cup products,lower central series,and holonomy Lie algebras |
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Authors: | Alexander I Suciu He Wang |
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Institution: | 1. Department of Mathematics, Northeastern University, Boston, MA 02115, USA;2. Department of Mathematics and Statistics, University of Nevada, Reno, NV 89557, USA |
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Abstract: | We generalize basic results relating the associated graded Lie algebra and the holonomy Lie algebra of a group, from finitely presented, commutator-relators groups to arbitrary finitely presented groups. Using the notion of “echelon presentation,” we give an explicit formula for the cup-product in the cohomology of a finite 2-complex. This yields an algorithm for computing the corresponding holonomy Lie algebra, based on a Magnus expansion method. As an application, we discuss issues of graded-formality, filtered-formality, 1-formality, and mildness. 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. |
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Keywords: | Primary 20F40 57M05 secondary 16A27 17B70 20F14 20J05 Lower central series Holonomy Lie algebra Magnus expansion Cohomology ring Link group One-relator group |
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