Mathematical Research Letters

Volume 20 (2013)

Number 6

Combinatorial methods for the twisted cohomology of Artin groups

Pages: 1157 – 1175



M. Salvetti (Dipartimento di Matematica, Università di Pisa, Italy)

A. Villa (Department of Mathematical Sciences, ISI Foundation, Torino, Italy)


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.

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