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 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 cupproduct 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.

Formality properties of finitely generated groups and Lie algebras

“…Once again, let G be a finitely generated group. The next lemma is known; for a proof and more details, we refer to [77]. Lemma 5.6 ([53,59]).…”

Chen ranks and resonance varieties of the upper McCool groups

“…where E = V and ker µ A is the ideal generated by ker µ A . We refer to [32,35,34] for full details of this construction, and further references and background.…”

“…This computation is an application of the method from [15, §15.1.1]. Since we already found a Gröbner basis G for B n = im(Ψ), formula (34) insures that we only need to compute the Hilbert series of the resulting monomial ideal, in ≻ (im(Ψ)) = in ≻ (G ) . Recall from Theorem 4.1 and Lemma 3.4 that…”

Homology, lower central series, and hyperplane arrangements

“…Following [8,18,23,29,43,44], we define h(G, k), the holonomy Lie algebra of G, as the quotient of the free Lie algebra on H 1 (G, k) by the Lie ideal generated by the image of the holonomy map, (h 2 ) * :…”

“…In [23], Markl and Papadima extended the definition of the holonomy Lie algebra to integral coefficients. Further in-depth studies were done by Papadima-Suciu [29] and Suciu-Wang [43,44]; in particular, the more general case when the group H = H 1 (G, Z) is allowed to have torsion is treated in [43].…”