Combinatorial methods for the twisted cohomology of Artin groups

Salvetti, Mario; Villa, Aleix Ruiz de

doi:10.4310/mrl.2013.v20.n6.a13

Cited by 10 publications

(19 citation statements)

References 11 publications

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“…They are described in Tables 7 and 8, where again we employ the notation {d} = R/(ϕ d ). We recover the results of [DCPSS99] (for the finite cases) and [SV13] (for the affine cases), except for minor corrections in the cases E 8 andẼ 8 .…”

Section: Exceptional Casessupporting

confidence: 81%

“…In this paper we study the local homology of Artin groups with coefficients in the Laurent polynomial ring R = Q[q ±1 ], where each standard generator acts as a multiplication by −q. This homology has been already thoroughly investigated for groups of finite type [Fre88,DCPSS99,DCPS01,Cal05,Sal15,PS18], also with integral coefficients [CS04,Cal06], and for some groups of affine type [CMS08a,CMS08b,CMS10,SV13,PS18]. This work is meant to be a natural continuation of [PS18], and is based on the combinatorial techniques developed in [SV13,PS18].…”

Section: Introductionmentioning

confidence: 99%

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On the local homology of Artin groups of finite and affine type

Paolini

2019

Algebr. Geom. Topol.

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We study the local homology of Artin groups using weighted discrete Morse theory. In all finite and affine cases, we are able to construct Morse matchings of a special type (we call them "precise matchings"). The existence of precise matchings implies that the homology has a square-free torsion. This property was known for Artin groups of finite type, but not in general for Artin groups of affine type. We also use the constructed matchings to compute the local homology in all exceptional cases, correcting some results in the literature.Theorem 3.1. Every Artin group of finite or affine type admits a ϕ-precise matching for each cyclotomic polynomial ϕ.Corollary 3.2. Let G W be an Artin group of finite or affine type. Then the local homology H * (X W ; R) has no ϕ k -torsion for k ≥ 2.We are able to use precise matchings to carry out explicit homology computations for all exceptional finite and affine cases. In particular we recover the results of [DCPSS99, SV13], with small corrections. The matchings we find for D n ,B n , and D n are quite complicated, so we prefer to omit explicit homology computations for these cases (the homology for D n andB n was already computed in [DCPSS99] and [CMS10], respectively). The remaining finite and affine cases, namely A n , B n ,Ã n andC n , were already discussed in [PS18].We also provide a software library which can be used to generate matchings for any finite or affine Artin group, check preciseness, and compute the homology. Source code and instructions are available online [Pao17].This paper is structured as follows. In Section 2 we review the general combinatorial framework developed in [SV13, PS18]. We introduce the local homology H * (X W ; R), which is the object of our study, together with algebraic complexes to compute it. We present weighted discrete Morse theory and precise matchings, and recall some useful results. In Section 3 we state and discuss the main results of this paper. Subsequent sections are devoted to the proof of the main theorem. In Section 4 we show that it is enough to construct precise matchings for irreducible Artin groups. In Section 5 we recall the computation of the weight of irreducible components of type A n , B n and D n , which is used later. In Sections 6-10 we construct precise matchings for the families A n , D n ,B n ,D n and I 2 (m). Finally, in Section 11 we deal with the exceptional cases. Local homology of Artin groups via discrete Morse theoryIn this section we are going to recall the general framework of [SV13, PS18] for the computation of the local homology H * (X W ; R).Let (W, S) be a Coxeter system on a finite generating set S, and let Γ be the corresponding Coxeter graph (with S as its vertex set). Denote by G W the corresponding Artin group, with standard generating set Σ = {g s | s ∈ S}. Define K W as the (finite) simplicial complex over S given by K W = {σ ⊆ S | the parabolic subgroup W σ generated by σ is finite}.

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Section: Exceptional Casessupporting

confidence: 81%

Section: Introductionmentioning

confidence: 99%

On the local homology of Artin groups of finite and affine type

Paolini

2019

Algebr. Geom. Topol.

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“…In this section we are going to recall some definitions and constructions from [SV13] (see also [MSV12]).…”

Section: Preliminariesmentioning

confidence: 99%

“…In [SV13] a more combinatorial method of calculation was introduced, based on the application of discrete Morse theory to a particular class of sheaves over posets (called weighted sheaves). It is interesting to notice that similar ideas were considered later in [CGN16], even if there the authors were mainly interested in computational aspects.…”

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confidence: 99%

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Weighted sheaves and homology of Artin groups

Paolini

Salvetti

2018

Algebr. Geom. Topol.

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In this paper 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 casẽ Cn, as well as for the cases An, Bn andÃn (which were already done before by different methods).

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Some Combinatorial Constructions and Relations with Artin Groups

Salvetti

2015

Springer INdAM Series

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Combinatorial methods for the twisted cohomology of Artin groups

Cited by 10 publications

References 11 publications

On the local homology of Artin groups of finite and affine type

On the local homology of Artin groups of finite and affine type

Weighted sheaves and homology of Artin groups

Some Combinatorial Constructions and Relations with Artin Groups

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