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

Abstract

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}.

Proof of the $$K(\pi , 1)$$ conjecture for affine Artin groups

“…The following are some consequences of the K (π, 1) conjecture: affine Artin groups are torsion-free (this already follows from the construction of McCammond and Sulway [49]); they have a classifying space with a finite number of cells (see [60]); the well studied homology and cohomology of Y W coincides with the homology and cohomology of the corresponding affine Artin group G W (see [19][20][21]55,57,61]); the natural map between the classifying space of an affine Artin monoid and the classifying space of the corresponding Artin group is a homotopy equivalence (see [33,34,52,53]).…”

La conjecture du $K(π,1)$ pour les groupes d'Artin affines (d'après Paolini et Salvetti)

“…L'un des intérêts de la conjecture du K(π, 1) est qu'elle permet de calculer l'homologie et la cohomologie du groupe d'Artin G W à partir de l'espace de configuration Y W , ce qui a déjà fait l'objet de nombreux travaux (Cohen, 1973 ;Salvetti, 1994 ;Charney et Davis, 1995a ;De Concini et Salvetti, 2000 ;Callegaro et Salvetti, 2004 ;Callegaro, Moroni et Salvetti, 2008 ;Paolini et Salvetti, 2018 ;Paolini, 2019b).…”

“…Cette approche duale pour la conjecture du K(π, 1) est également présentée de manière synthétique dans (Paolini, 2021). En particulier, il est envisageable que cette stratégie puisse fournir une preuve de la conjecture du K(π, 1) pour d'autres groupes d'Artin : le cas des groupes d'Artin de rang 3 est annoncé dans (Paolini, 2021, Theorem 6.1).…”

Weighted sheaves and homology of Artin groups

“…Future works will focus on other families of Artin groups. In particular it seems that precise matchings can be constructed in all finite and affine cases (see [Pao17]), possibly allowing explicit homology computations.…”

“…In a recent preprint [Pao17], the first author succeeded to construct precise matchings for all remaining Artin groups of finite and affine type. Therefore precise matchings seem to be a suitable and effective tool for studying the homology of Artin groups.…”