Volume 18, issue 7 (2018)

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Weighted sheaves and homology of Artin groups

Giovanni Paolini and Mario Salvetti

Algebraic & Geometric Topology 18 (2018) 3943–4000
Abstract

We expand the theory of weighted sheaves over posets, and use it to study the local homology of Artin groups. First, we use such theory to relate the homology of classical braid groups with the homology of certain independence complexes of graphs. Then, in the context of discrete Morse theory on weighted sheaves, we introduce a particular class of acyclic matchings. Explicit formulas for the homology of the corresponding Morse complexes are given, in terms of the ranks of the associated incidence matrices. We use such method to perform explicit computations for the new affine case ${\stackrel{̃}{C}}_{n}$, as well as for the cases ${A}_{n}$, ${B}_{n}$ and ${Ã}_{n}$ (which were already done before by different methods).

Keywords
Artin groups, hyperplane arrangements, discrete Morse theory
Mathematical Subject Classification 2010
Primary: 05E45, 20F36, 52C35