Mathematical Research Letters
Volume 20 (2013)
Combinatorial methods for the twisted cohomology of Artin groups
Pages: 1157 – 1175
In this paper we introduce certain “combinatorial sheaves” on posets, which we call weighted sheaves, and we relate their cohomology, computed by a weighted complex, with the twisted cohomology of Artin groups. It turns out that to each Artin group one can associate a weighted sheaf, where the poset is given by the simplicial complex of all finite parabolic subgroups, and the cohomology of the Artin group with coefficients in a module of Laurent polynomials (interesting for geometrical reasons) is computed by the associated weighted complex. We connect this theory with the so called “Discrete Morse Theory”: a weighted matching on the weighted complex gives rise to a Morse complex computing the cohomology. We give a natural filtration of the weighted complex, which is compatible with the weighted matching, so obtaining a converging spectral sequence. We use such machinery to compute the twisted conomology for all exceptional type affine Artin groups.