• local homology
• braid arrangement
• algebraic topology
• Discrete morse theory
• arrangement
• hyperplane

The aim of this thesis is to study the complement of a hyperplane arrangement using the techniques of Discrete Morse theory.

A hyperplane arrangement is simply a set A={H_1,H_2,...} of hyperplanes in a vector space. This object has been widely studied especially to find correlations between its topological properties and the combinatorics of the intersections of the hyperplanes. One of the most studied topological objects is the complement, i.e. the vector space minus the hyperplanes and its homology and homotopy groups. One of the question that we are going to answer is if this complement is a minimal space, meaning that it is homotopy equivalent to a CW-complex with as many i-cells as the i-th Betti number. We will see that the answer is positive in various settings.

Having a minimal complex, if it is given explicitly, can also help in studying various properties of the complement, we will focus in particular on abelian local homology. Local homology is an important tool for the study of hyperplane arrangements because it gives us informations on a special fibration on the complement, called Milnor fibration as well as informations about the characteristic varieties. Even if there is plenty of research on the subject there is still a lot unknown about local homology even for the most famous arrangements, like the Braid ones.

In the first chapter we talk about Discrete Morse theory, first introduced by Forman. The aim is to reduce a CW-complex or in general various type of topological and combinatorial object to smaller ones, called Morse complex with a series of elementary collapsments such that the properties of the complex are still the same. We will study explore the correlation of this theory with shellability and focus on different aspect, all of which will became useful in the following chapters.

In the second chapter, we first give a brief introduction to the theory of hyperplane arrangement, presenting some of the most important known results, concerning in particular their combinatorial properties and homology groups.
In the special case of complexified real arrangement we introduce the Salvetti complex, a $CW-$complex homotopy equivalent to the complementary of the arrangement. We then review three different articles that with similar techniques have reduced this complex to a minimal one (with as many cells as the Betti numbers).

The following chapter introduce the concept of local homology and its correlations with Discrete Morse theory, in particular how we could compute the local homology of the Morse complex. We focus then our attention to a special kind of hyperplane arrangement, called the Braid arrangement and we explicitly write a program in Sage to compute the boundary in local homology.

In the last chapter we try to give our small contribute to the subject. It is a joint work with Giovanni Paolini in which in a similar way to what has been done by Delucchi in the case of oriented matroids, we reduce the Salvetti complex in the case of affine, locally finite hyperplane arrangment to a minimal Morse complex giving a special characterization to the critical cells.