We prove the $$K(\pi ,1)$$ K ( π , 1 ) conjecture for affine Artin groups: the complexified complement of an affine reflection arrangement is a classifying space. This is a long-standing problem, due to Arnol’d, Pham, and Thom. Our proof is based on recent advancements in the theory of dual Coxeter and Artin groups, as well as on several new results and constructions. In particular: we show that all affine noncrossing partition posets are EL-shellable; we use these posets to construct finite classifying spaces for dual affine Artin groups; we introduce new CW models for the orbit configuration spaces associated with arbitrary Coxeter groups; we construct finite classifying spaces for the braided crystallographic groups introduced by McCammond and Sulway.

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).

We introduce an asymmetric distance in the space of learning tasks and a framework to compute their complexity. These concepts are foundational for the practice of transfer learning, whereby a parametric model is pre-trained for a task, and then fine tuned for another. The framework we develop is non-asymptotic, captures the finite nature of the training dataset and allows distinguishing learning from memorization. It encompasses, as special cases, classical notions from Kolmogorov complexity and Shannon and Fisher information. However, unlike some of those frameworks, it can be applied to large-scale models and real-world datasets. Our framework is the first to measure complexity in a way that accounts for the effect of the optimization scheme, which is critical in deep learning.

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}.

A theorem proved by Dobrinskaya in 2006 shows that there is a strong connection between the K(π, 1) conjecture for Artin groups and the classifying spaces of Artin monoids. More recently Ozornova obtained a different proof of Dobrinskaya's theorem based on the application of discrete Morse theory to the standard CW model of the classifying space of an Artin monoid. In Ozornova's work there are hints at some deeper connections between the above-mentioned CW model and the Salvetti complex, a CW complex which arises in the combinatorial study of Artin groups. In this work we show that such connections actually exist, and as a consequence we derive yet another proof of Dobrinskaya's theorem.