• subspace arrangements
• minimality
• hyperplane arrangements
• discrete Morse theory
• configuration spaces

The theory of Hyperplane Arrangements (more generally, Subspace
Arrangements) is developing in the last (at least) three decades as an
interesting part of Mathematics, which derives from and at the same
time connects different classical branches. Among them we have: the
theory of root systems (so, indirectly, Lie theory); Singularity
theory, by the classical connection with simple singularities and
braid groups and related groups (Artin groups); Combinatorics, through
for example Matroid and Oriented Matroid theory; Algebraic Geometry,
in connection with certain moduli spaces of genus zero curves and also
through the classical study of the topology of Hypersurface
complements; the theory of Generalized Hypergeometric Functions, and
the connected development of the study of \emph{local system}
cohomologies; recently, the theory of box splines, partition
functions, index theory.

Most of the theory is spread into a big number of papers, but there
exists also (few) dedicated books, or parts of books, as
\cite{goresky_mcpherson}, \cite{orlik_terao}, and the recent book
\cite{deconcini_procesi}.

The subject of this thesis concerns some topological aspects of the
theory which we are going to outline here.

So, consider an hyperplane arrangement $\mathcal A$ in $\R^n.$ We
assume here that $\mathcal A$ is finite, but most of the results hold
with few modifications for any affine (locally finite) arrangement. It
was known by general theories that the complement to the complexified
arrangement $\mathcal M(\mathcal A)$ has the homotopy type of an
$n-$dimensional complex, and in \cite{salvetti87} an explicit
construction of a combinatorial complex (denoted since then as the
Salvetti complex, here denoted by $\S$) was made. In general, such
complex has more $k-$cells than the $k$-th Betti number of $\mathcal
M(\mathcal A).$ It has been known for a long time that the cohomology
of the latter space is free, and a combinatorial description of such
cohomology was found (see \cite{orlik_terao} for references). The
topological type of the
complement is not combinatorial for general arrangements, but it is
still unclear if this is the case for special classes of
arrangements. Nevertheless, suspecting special properties for the
topology of the complement, it was proven that the latter enjoys a
strong \emph{minimality} condition. In fact, in
\cite{dimca_papadima},\cite{randell} it was shown that $\mathcal
M(\mathcal A)$ has the homotopy type of a $CW$-complex having exactly
$\beta_k$ $k$-cells, where $\beta_k$ is the $k$-th Betti number.

This was an \emph{existence-type} result, with no explicit description
of the minimal complex.

A more precise description of the minimal complex, in the case of real
defined arrangements, was found in \cite{yoshinaga}, using classical
Morse theory. A better explicit description was found in
\cite{salvsett}, where the authors used Discrete Morse theory over $\S$ (as
introduced in \cite{forman, forman1}). There they introduce a
\emph{total} ordering (denoted \emph{polar ordering}) for the set of
\emph{facets} of the induced stratification of $\R^n,$ and define an
explicit discrete vector field over the face-poset of $\S$. There are
as many \emph{$k$-critical cells} for this vector field as the $k-$th
Betti number ($k\geq 0$). It follows from discrete Morse theory that
such a discrete vector field produces:

i) a homotopy equivalence of $\S$ with a minimal complex;

ii) an explicit description (up to homotopy) of the boundary maps of
the minimal complex, in terms of \emph{alternating paths}, which can
be computed explicitly from the field.

A different construction (which has more combinatorial flavor) was
given in \cite{delucchi} (see also \cite{delucchi_settepanella}).


In this thesis we consider this kind of topological problems around
minimality. First, even if the above construction allows in theory to
produce the minimal complex explicitly, the boundary maps that one
obtains by using the alternating paths are not \emph{themselves
minimal,} in the sense that several pairs of the same critical
cell can delete each other inside the attaching maps of the bigger
dimensional critical cells. So, a problem is to produce a minimal complex with
\emph{minimal} attaching maps.

We are able to do that in the two-dimensional affine case (see chapter
\ref{sec:formula}, \cite{gaiffimorisalvetti}).

Next, we generalize the construction of the vector field to the case
of so called \emph{$d$-complexified} arrangements.

First, consider classical Configuration Spaces in $\R^d$ (sometimes
written as $F(n,\R^d)$) : they are defined as the set of ordered
$n-$tuples of \emph{pairwise different} points in $\R^d.$ Taking
coordinates in $(\R^d)^n=\R^{nd}$
$$x_{ij},\ i=1,\dots,n,\ j=1,\dots,d,$$
one has
$$F(n,\R^d)\ =\ \R^{nd}\setminus\cup_{i\neq j}\ H_{ij}^{(d)},$$
where $H_{ij}^{(d)}$ is the codimension $d$-subspace
$$\cap_{k=1,\dots,d}\ \{x_{ik}=x_{jk}\}.$$
So, the latter subspace is the intersection of $d$ hyperplanes in
$\R^{nd},$ each obtained by the hyperplane
$H_{ij}=\{x\in\R^n\ :\ x_i=x_j\},$ considered on the $k-$th component
in $(\R^n)^d=\R^{nd},$ $k=1,\dots,d.$

By a \emph{Generalized Configuration Space} (for brevity, simply a
Configuration Space) we mean an analog construction, which starts from
any \emph{Hyperplane Arrangement} $\A$ in $\R^n$. For each $d>0,$ one
has a\ $d-$\emph{complexification} \
$\A^{(d)}\subset M^d$ of $\A,$ which is given by the collection $
\{H^{(d)},\ H\in\A\}$ of the \emph{$d$-complexified} subspaces. The
\emph{configuration space} associated to $\A$ is the complement to the
subspace arrangement
$$ \M^{(d)}\ =\ \M(\A)^{(d)} :=\ (\R^n)^d \setminus \bigcup_{H\in
\A} H^{(d)}\ .$$
For $d=2$ one has the standard complexification of a real hyperplane
arrangement. There is a natural inclusion $\M^{(d)}\hookrightarrow
\M^{(d+1)}$ and the limit space is contractible (in case of an
arrangement associated to a reflection group $W,$ the limit of the
orbit space with respect to the action of $W$ gives the classifying
space of $W;$ see \cite{deconcini_salvetti00}) .

In this thesis we give an explicit construction of a minimal CW-complex for the
configuration space $\M(\A)^{(d)},$ for all $d\geq 1.$
That is, we explicitly produce a $CW$-complex having as many
$i$-cells as the $i$-th Betti number $\beta_i$ of $\M(\A)^{(d)},$ $i\geq
0$.

For $d=1$ the result is trivial, since $\M^{(1)}$ is a disjoint union
of convex sets (the \emph{chambers}). Case $d=2$ was discussed
above.
For $d>2$ the configuration spaces are simply-connected, so by general
results they have the homotopy type of a minimal
$CW$-complex. Nevertheless, having explicit "combinatorial" complexes
is useful in order to produce geometric bases for the cohomology. In
fact, we give explicit bases for the homology (and cohomology) of
$\Md{d+1}$ which we call ($d$)-\emph{polar bases}.
As far as we know, there is no other precise description of a
geometric $\Z$-basis in the literature, except for some particular
arrangements, in spite of the fact that the $\Z$-module structure of
the homology is well known: it derives from a well known formula in
\cite{goresky_mcpherson} that such homology depends only on the
intersection lattice of the $d$-complexification $\A^{(d)},$ and such
lattice is the same for all $d\geq 1.$
The tool we use here is still discrete Morse theory. Starting from the
previous explicit construction in \cite{deconcini_salvetti00} of a
non-minimal $CW$-complex (see also \cite{bjorner_ziegler}) which we
denote here by $\S^{(d)},$ which has the homotopy type of
$\M^{(d+1)},$ we construct an explicit \emph{combinatorial gradient
vector field} on $\S^{(d)}$ and we give a precise description of the
critical cells. One finds that critical cells live in dimension $id,$
for $i=1,\dots,n',$ where $n'$ is the \emph{rank} of the arrangement
$\A$ ($n'\leq n$).

Notice that the proof of minimality, in case $d>2,$ is straightforward
from our construction because of the gap between the dimensions of the
critical cells.


One can conjecture that \emph{torsion-free subspace arrangements are
minimal}: that is, when the complement of the arrangement has
torsion-free cohomology, then it is a minimal space.

We pass now to a more precise description of the contents of the
several parts of the thesis.

Chapters \ref{prerequisiti}, \ref{sottospazi} and
\ref{salvettisettepanella} are introductive, the original part can be
found at most in chapters \ref{sec:formula} and \ref{configuration}.

Chapter \ref{prerequisiti} is an introductory collection of the main
tools needed in the following parts. It includes: Orlik-Solomon algebra
and related topics, as the so called \emph{broken circuit bases}; the
definition of Salvetti complex; the main definitions and results of
the Discrete Morse Theory, following the original work by Forman
(\cite{forman,forman1}).


In chapter \ref{sottospazi} we deal with general
subspace arrangements. In section \ref{Gorformula} we recall
Goresky-MacPherson formula. We consider here the explicit example
given in \cite{jewell} of a subspace arrangement such that its
complement is not torsion-free. This arrangement is composed with six
codimensional-5 coordinate subspaces in $\R^{10}$ (we make complete
computation of the cohomology of the complement by using
Goresky-MacPherson formula).


In section \ref{spaziconfigurazione} we define generalized
$d-$configuration spaces $\mathcal{M}(\A)^{(d)}$,
and the generalized Salvetti complex
$\S^{(d)},$ whose cells correspond to all \emph{chains}
$(C\<F_1\<\dots\<F_d),$ where $C$ is a chamber and the $F_i$'s are
facets of the induced stratification $\Fi(\A)$ of $\R^n$ (and $\<$ is the
standard face-ordering in $\Fi(\A)$).

In chapter \ref{salvettisettepanella} we present the reduction of the
complex $\S=\S^{(1)}$ using discrete Morse theory, following \cite{salvsett}.
We define a system of polar coordinates in $\R^n$, and the induced
polar ordering on the stratification
$\Fi(\A).$ Next,
we define a gradient vector field $\Gamma$ on the set
of cells of $\S$; the critical cells of $\Gamma$ are in one-to-one
correspondence with the cells of a new $CW$-complex, which has the same
homotopy type as $\S.$ One can verify that the number of critical cells
of dimension $k$ equals the $k-$th Betti
number, so the latter $CW$-complex is minimal.


The main original part of our thesis is contained in the last two chapters.

In chapter \ref{sec:formula} we consider the two-dimensional case. For
any affine line arrangement $\A,$ we give explicit \emph{minimal} attaching
maps for the minimal two-complex corresponding to the polar gradient
vector field. After considering the central case, the proof is by
induction on the number of $0$-dimensional facets of $\A.$

Of course, presentations of the fundamental
group of the complement follow straightforward from these explicit
boundary formulas.


In chapter \ref{configuration} we apply discrete Morse theory to the
complex $\S^{(d)}$. Even if the philosophy here is similar to that used
for $d=1$, the extension to the case $d>1$ is not trivial. To
construct a gradient field on $\S^{(d)},$ we have to consider on the
$i$th-component of the chains $(C\<F_1\<\dots\<F_d)\in \Sd$
either the polar ordering which is
induced on the arrangement "centered" in the $(i+1)$th-component of
the chain, or the opposite of such ordering, according to the parity
of $d-i.$ Then we use a double induction over $d$ and the dimension of
a sub-arrangement of $\A.$


Several examples are considered in order to better illustrate our results.