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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 C̃n, as well as for the cases An, Bn and Ãn (which were already done before by different methods).
Keywords: Artin groups, hyperplane arrangements, discrete Morse theory
Paolini, Giovanni 1 ; Salvetti, Mario 2
@article{10_2140_agt_2018_18_3943,
author = {Paolini, Giovanni and Salvetti, Mario},
title = {Weighted sheaves and homology of {Artin} groups},
journal = {Algebraic and Geometric Topology},
pages = {3943--4000},
publisher = {mathdoc},
volume = {18},
number = {7},
year = {2018},
doi = {10.2140/agt.2018.18.3943},
url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2018.18.3943/}
}
TY - JOUR AU - Paolini, Giovanni AU - Salvetti, Mario TI - Weighted sheaves and homology of Artin groups JO - Algebraic and Geometric Topology PY - 2018 SP - 3943 EP - 4000 VL - 18 IS - 7 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.2140/agt.2018.18.3943/ DO - 10.2140/agt.2018.18.3943 ID - 10_2140_agt_2018_18_3943 ER -
%0 Journal Article %A Paolini, Giovanni %A Salvetti, Mario %T Weighted sheaves and homology of Artin groups %J Algebraic and Geometric Topology %D 2018 %P 3943-4000 %V 18 %N 7 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.2140/agt.2018.18.3943/ %R 10.2140/agt.2018.18.3943 %F 10_2140_agt_2018_18_3943
Paolini, Giovanni; Salvetti, Mario. Weighted sheaves and homology of Artin groups. Algebraic and Geometric Topology, Tome 18 (2018) no. 7, pp. 3943-4000. doi: 10.2140/agt.2018.18.3943
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