Minimality of hyperplane arrangements and configuration spaces: a combinatorial approach

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.