Lower central series and free resolutions of hyperplane arrangements

Abstract

Abstract. If M is the complement of a hyperplane arrangement, and A = H * (M, k) is the cohomology ring of M over a field k of characteristic 0, then the ranks, φ k , of the lower central series quotients of π 1 (M ) can be computed from the Betti numbers, b ii = dim Tor A i (k, k) i , of the linear strand in a minimal free resolution of k over A. We use the Cartan-Eilenberg change of rings spectral sequence to relate these numbers to the graded Betti numbers,From this analysis, we recover a formula of Falk for φ 3 , and obtain a new formula for φ 4 . The exact sequence of low-degree terms in the spectral sequence allows us to answer a question of Falk on graphic arrangements, and also shows that for these arrangements, the algebra A is Koszul if and only if the arrangement is supersolvable.We also give combinatorial lower bounds on the Betti numbers, b i,i+1 , of the linear strand of the free resolution of A over E; if the lower bound is attained for i = 2, then it is attained for all i ≥ 2. For such arrangements, we compute the entire linear strand of the resolution, and we prove that all components of the first resonance variety of A are local. For graphic arrangements (which do not attain the lower bound, unless they have no braid subarrangements), we show that b i,i+1 is determined by the number of triangles and K 4 subgraphs in the graph.

The Pure Braid Groups and Their Relatives

“…where c m is the number of m-dimensional components of R 1 (G). Much work has gone into proving this conjecture, with special cases being verified in [63,58,65].…”

“…The Chen ranks of the pure braid groups P n were computed in [24], while an explicit relation between the Chen ranks and the resonance varieties of an arrangement group was conjectured in [67]. Building on work from [26,63,65] and especially [58], Cohen and Schenck confirmed this conjecture in [23] for a class of 1-formal groups which includes arrangement groups. In the process, they also computed the Chen ranks θ k (wP n ) for k sufficiently large.…”

The multi-modal Australian ScienceS Imaging and Visualization Environment (MASSIVE) high performance computing infrastructure: applications in neuroscience and neuroinformatics research

“…Both M1 and M2 have a GPFS (Schmuck and Haskin, 2002) file system that is capable of a combined 5 GB+ per second write speed. This capability has proved essential to support both the fast capture of data from instruments, and file system intensive image processing workloads.…”

G-Quadratic, LG-Quadratic, and Koszul Quotients of Exterior Algebras

“…It is an open question whether there are Koszul Orlik-Solomon algebras arising from non-supersolvable arrangements; see [31,Problem 82] and [33, p. 487]. The two notions are equivalent for hypersolvable arrangements [20], graphic arrangements [32], Dirichlet arrangements [24], root ideal arrangements [18], and others [36].…”