Mario Salvetti

Combinatorial Morse theory and minimality of hyperplane arrangements

which is generic with respect to the arrangement, ie Vi intersects transversally all codimension‐i intersections of hyperplanes. The interesting main result is a correspondence between the k ‐cells of the minimal complex and the set of chambers which intersect Vk but do not intersect Vk 1 . The arguments still use the Morse theoretic proof of the Lefschetz theorem, and some analysis of the critical cells is given. Unfortunately, the description does not allow one to understand exactly the attaching maps of the cells of a minimal complex.

On the homotopy theory of complexes associated to metrical-hemisphere complexes

Abstract In the first section we introduce a certain class of sellular complexes, called metrical- hemisphere complexes (MH-complexes), which generalize oriented matroids, and study their main properties. In Section 2 another cellular complex is associated to each MH-complex; this construction generalizes that given in a preceding work. In Section 3 homotopy theory of the associated complexes is studied by introducing a map between a certain category of positive paths and the fundamental grupoid. If J is injective then the word problem for the fundamental group is solvable. In Sections 4, 5 higher homotopy groups are considered, giving sufficient criterions for the k(π, 1) property. Also, the results by Deligne concerning the simplicial arrangements is reproved in larger generality. In Section 6 the preceding results are exploited to produce newk(π, 1) spaces associated to arrangement of pseudo-hemispheres.

The K(p,1) problem for the affine Artin group of type Bn and its cohomology

In this paper we prove that the complement to the affine complex arrangement of type \widetilde{B}_n is a K(\pi, 1) space. We also compute the cohomology of the affine Artin group G of type \widetilde{B}_n with coefficients over several interesting local systems. In particular, we consider the module Q[q^{\pm 1}, t^{\pm 1}], where the first n-standard generators of G act by (-q)-multiplication while the last generator acts by (-t)-multiplication. Such representation generalizes the analog 1-parameter representation related to the bundle structure over the complement to the discriminant hypersurface, endowed with the monodromy action of the associated Milnor fibre. The cohomology of G with trivial coefficients is derived from the previous one.

Cohomology of affine Artin groups and applications

The result of this paper is the determination of the cohomology of Artin groups of type An, B n and A n with non-trivial local coefficients. The main result is an explicit computation of the cohomology of the Artin group of type B n with coefficients over the module Q[q ±1 ,t ±1 ]. Here the first n - 1 standard generators of the group act by (-q)-multiplication, while the last one acts by (-t)-multiplication. The proof uses some technical results from previous papers plus computations over a suitable spectral sequence. The remaining cases follow from an application of Shapiros lemma, by considering some well-known inclusions: we obtain the rational cohomology of the Artin group of affine type An as well as the cohomology of the classical braid group Br n with coefficients in the n-dimensional representation presented in Tong, Yang, and Ma (1996). The topological counterpart is the explicit construction of finite CW-complexes endowed with a free action of the Artin groups, which are known to be K(π, 1) spaces in some cases (including finite type groups). Particularly simple formulas for the Euler-characteristic of these orbit spaces are derived.

Artin groups associated to infinite Coxeter groups

Abstract First we remark that the cellular complex constructed by Salvetti (1994) can be considered as a ‘topological’ invariant of a graph, so its cohomology is also an invariant. We use the construction of Salvetti (1994) to calculate the cohomology of the Artin group associated to the complete graph K n , using coefficients in a local system over Z [ q , q −1 ]. The standard cohomology over Z is obtained by specializing q to 1. While doing such computations, we obtain also an explicit rational function for the Poincare series of the Coxeter group associated to K π , and note that it has exponential growth for n ⩾4.

Heart deformation pattern analysis through shape modelling

In this paper, we present an approach to the description of time-varying anatomical structures. The main goal is to compactly but faithfully describe the whole heart cycle in such a way to allow for deformation pattern characterization and assessment. Using such an encoding, a reference database can be built, thus permitting similarity searches or data mining procedures.

Multi-scale Representation and Persistency for Shape Description

Extraction, organization and exploitation of topological features are emerging topics in computer vision and graphics. However, such kind of features often exhibits weak robustness with respect to small perturbations and it is often unclear how to distinguish truly topological features from topological noise. In this paper, we present an introduction to persistence theory, which aims at analyzing multi-scale representations from a topological point of view. Besides, we extend the ideas of persistency to a more general setting by defining a set of discrete invariants.

Some topological problems on the configuration spaces of Artin and Coxeter groups

In the first part we review some topological and algebraic aspects in the theory of Artin and Coxeter groups, both in the finite and infinite case (but still, finitely generated). In the following parts, among other things, we compute the Schwartz genus of the covering associated to the orbit space for all affine Artin groups. We also give a partial computation of the cohomology of the braid group with non-abelian coefficients coming from geometric representations. We introduce an interesting class of “sheaves over posets”, which we call “weighted sheaves over posets”, and use them for explicit computations.

Cohomology of Coxeter groups

Abstract Let W be a finitely generated Coxeter group. We describe a method which is useful in computing the cohomology of W with local coefficients. It is based on the determination of an explicit combinatorial resolution of Z over Z [ W ] (developed in a previous paper) which grows polynomially with the dimension. Let S be a Coxeter set of generators for W : in dimension k , the rank of the resolution equals the number of flags Γ 1 ⊃Γ 2 ⊃⋯, Γ i ⊂S , having cardinality k , while the coboundary is explicitly given in terms of minimal coset representatives of W . Here we use such a method in order to produce some cohomologies in the cases of dihedral groups and symmetric groups.

On the Twisted Cohomology of Affine Line Arrangements

Let \(\mathcal{A}\) be an affine line arrangement in \(\mathbb{C}^{2}\), with complement \(\mathcal{M}(\mathcal{A})\). The twisted (co)homology of \(\mathcal{M}(\mathcal{A})\) is an interesting object which has been considered by many authors. In this paper we give a vanishing conjecture of a different nature with respect to the known results: namely, we conjecture that if the graph of double points of the arrangement is connected then there is no nontrivial monodromy. This conjecture is obviously combinatorial (meaning that it depends only on the lattice of the intersections). We prove it in some cases with stronger hypotheses. We also consider the integral case, relating the property of having trivial monodromy over \(\mathbb{Z}\) with a certain property of “commutativity” of the fundamental group up to some subgroup. At the end, we give several examples and computations.