## Tesi etd-08242010-120426 |

Thesis type

Tesi di dottorato di ricerca

Author

MORI, FRANCESCA

URN

etd-08242010-120426

Title

Minimality of hyperplane arrangements and configuration spaces: a combinatorial approach

Settore scientifico disciplinare

MAT/03

Corso di studi

MATEMATICA

Commissione

**tutor**Salvetti, Mario

Parole chiave

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

Data inizio appello

15/09/2010;

Consultabilità

completa

Riassunto analitico

The theory of Hyperplane Arrangements (more generally, Subspace<br>Arrangements) is developing in the last (at least) three decades as an<br>interesting part of Mathematics, which derives from and at the same<br>time connects different classical branches. Among them we have: the<br>theory of root systems (so, indirectly, Lie theory); Singularity<br>theory, by the classical connection with simple singularities and<br>braid groups and related groups (Artin groups); Combinatorics, through<br>for example Matroid and Oriented Matroid theory; Algebraic Geometry,<br>in connection with certain moduli spaces of genus zero curves and also<br>through the classical study of the topology of Hypersurface<br>complements; the theory of Generalized Hypergeometric Functions, and<br>the connected development of the study of \emph{local system}<br>cohomologies; recently, the theory of box splines, partition<br>functions, index theory. <br><br>Most of the theory is spread into a big number of papers, but there<br>exists also (few) dedicated books, or parts of books, as<br>\cite{goresky_mcpherson}, \cite{orlik_terao}, and the recent book<br>\cite{deconcini_procesi}. <br><br>The subject of this thesis concerns some topological aspects of the<br>theory which we are going to outline here. <br><br>So, consider an hyperplane arrangement $\mathcal A$ in $\R^n.$ We<br>assume here that $\mathcal A$ is finite, but most of the results hold<br>with few modifications for any affine (locally finite) arrangement. It<br>was known by general theories that the complement to the complexified<br>arrangement $\mathcal M(\mathcal A)$ has the homotopy type of an<br>$n-$dimensional complex, and in \cite{salvetti87} an explicit<br>construction of a combinatorial complex (denoted since then as the<br>Salvetti complex, here denoted by $\S$) was made. In general, such<br>complex has more $k-$cells than the $k$-th Betti number of $\mathcal<br>M(\mathcal A).$ It has been known for a long time that the cohomology<br>of the latter space is free, and a combinatorial description of such<br>cohomology was found (see \cite{orlik_terao} for references). The<br>topological type of the <br>complement is not combinatorial for general arrangements, but it is<br>still unclear if this is the case for special classes of<br>arrangements. Nevertheless, suspecting special properties for the<br>topology of the complement, it was proven that the latter enjoys a<br>strong \emph{minimality} condition. In fact, in<br>\cite{dimca_papadima},\cite{randell} it was shown that $\mathcal<br>M(\mathcal A)$ has the homotopy type of a $CW$-complex having exactly<br>$\beta_k$ $k$-cells, where $\beta_k$ is the $k$-th Betti number. <br><br>This was an \emph{existence-type} result, with no explicit description<br>of the minimal complex. <br><br>A more precise description of the minimal complex, in the case of real<br>defined arrangements, was found in \cite{yoshinaga}, using classical<br>Morse theory. A better explicit description was found in<br>\cite{salvsett}, where the authors used Discrete Morse theory over $\S$ (as<br>introduced in \cite{forman, forman1}). There they introduce a<br>\emph{total} ordering (denoted \emph{polar ordering}) for the set of<br>\emph{facets} of the induced stratification of $\R^n,$ and define an<br>explicit discrete vector field over the face-poset of $\S$. There are<br>as many \emph{$k$-critical cells} for this vector field as the $k-$th<br>Betti number ($k\geq 0$). It follows from discrete Morse theory that<br>such a discrete vector field produces: <br> <br> i) a homotopy equivalence of $\S$ with a minimal complex; <br><br> ii) an explicit description (up to homotopy) of the boundary maps of<br> the minimal complex, in terms of \emph{alternating paths}, which can<br> be computed explicitly from the field. <br> <br>A different construction (which has more combinatorial flavor) was<br>given in \cite{delucchi} (see also \cite{delucchi_settepanella}). <br><br><br>In this thesis we consider this kind of topological problems around<br>minimality. First, even if the above construction allows in theory to<br>produce the minimal complex explicitly, the boundary maps that one<br>obtains by using the alternating paths are not \emph{themselves<br> minimal,} in the sense that several pairs of the same critical<br>cell can delete each other inside the attaching maps of the bigger<br>dimensional critical cells. So, a problem is to produce a minimal complex with<br>\emph{minimal} attaching maps. <br><br>We are able to do that in the two-dimensional affine case (see chapter<br>\ref{sec:formula}, \cite{gaiffimorisalvetti}). <br> <br>Next, we generalize the construction of the vector field to the case<br>of so called \emph{$d$-complexified} arrangements. <br><br>First, consider classical Configuration Spaces in $\R^d$ (sometimes<br>written as $F(n,\R^d)$) : they are defined as the set of ordered<br>$n-$tuples of \emph{pairwise different} points in $\R^d.$ Taking<br>coordinates in $(\R^d)^n=\R^{nd}$ <br>$$x_{ij},\ i=1,\dots,n,\ j=1,\dots,d,$$ <br>one has<br>$$F(n,\R^d)\ =\ \R^{nd}\setminus\cup_{i\neq j}\ H_{ij}^{(d)},$$<br>where $H_{ij}^{(d)}$ is the codimension $d$-subspace<br>$$\cap_{k=1,\dots,d}\ \{x_{ik}=x_{jk}\}.$$<br>So, the latter subspace is the intersection of $d$ hyperplanes in<br>$\R^{nd},$ each obtained by the hyperplane<br>$H_{ij}=\{x\in\R^n\ :\ x_i=x_j\},$ considered on the $k-$th component<br>in $(\R^n)^d=\R^{nd},$ $k=1,\dots,d.$ <br><br>By a \emph{Generalized Configuration Space} (for brevity, simply a<br>Configuration Space) we mean an analog construction, which starts from<br>any \emph{Hyperplane Arrangement} $\A$ in $\R^n$. For each $d>0,$ one<br>has a\ $d-$\emph{complexification} \ <br>$\A^{(d)}\subset M^d$ of $\A,$ which is given by the collection $<br>\{H^{(d)},\ H\in\A\}$ of the \emph{$d$-complexified} subspaces. The<br>\emph{configuration space} associated to $\A$ is the complement to the<br>subspace arrangement <br>$$ \M^{(d)}\ =\ \M(\A)^{(d)} :=\ (\R^n)^d \setminus \bigcup_{H\in<br> \A} H^{(d)}\ .$$ <br>For $d=2$ one has the standard complexification of a real hyperplane<br>arrangement. There is a natural inclusion $\M^{(d)}\hookrightarrow<br>\M^{(d+1)}$ and the limit space is contractible (in case of an<br>arrangement associated to a reflection group $W,$ the limit of the<br>orbit space with respect to the action of $W$ gives the classifying<br>space of $W;$ see \cite{deconcini_salvetti00}) .<br><br>In this thesis we give an explicit construction of a minimal CW-complex for the<br>configuration space $\M(\A)^{(d)},$ for all $d\geq 1.$<br>That is, we explicitly produce a $CW$-complex having as many<br>$i$-cells as the $i$-th Betti number $\beta_i$ of $\M(\A)^{(d)},$ $i\geq<br>0$. <br><br>For $d=1$ the result is trivial, since $\M^{(1)}$ is a disjoint union<br>of convex sets (the \emph{chambers}). Case $d=2$ was discussed<br>above. <br>For $d>2$ the configuration spaces are simply-connected, so by general<br>results they have the homotopy type of a minimal<br>$CW$-complex. Nevertheless, having explicit "combinatorial" complexes<br>is useful in order to produce geometric bases for the cohomology. In<br>fact, we give explicit bases for the homology (and cohomology) of<br>$\Md{d+1}$ which we call ($d$)-\emph{polar bases}. <br>As far as we know, there is no other precise description of a<br>geometric $\Z$-basis in the literature, except for some particular<br>arrangements, in spite of the fact that the $\Z$-module structure of<br>the homology is well known: it derives from a well known formula in<br>\cite{goresky_mcpherson} that such homology depends only on the<br>intersection lattice of the $d$-complexification $\A^{(d)},$ and such<br>lattice is the same for all $d\geq 1.$ <br>The tool we use here is still discrete Morse theory. Starting from the<br>previous explicit construction in \cite{deconcini_salvetti00} of a<br>non-minimal $CW$-complex (see also \cite{bjorner_ziegler}) which we<br>denote here by $\S^{(d)},$ which has the homotopy type of<br>$\M^{(d+1)},$ we construct an explicit \emph{combinatorial gradient<br> vector field} on $\S^{(d)}$ and we give a precise description of the<br>critical cells. One finds that critical cells live in dimension $id,$<br>for $i=1,\dots,n',$ where $n'$ is the \emph{rank} of the arrangement<br>$\A$ ($n'\leq n$). <br><br>Notice that the proof of minimality, in case $d>2,$ is straightforward<br>from our construction because of the gap between the dimensions of the<br>critical cells. <br><br><br>One can conjecture that \emph{torsion-free subspace arrangements are<br> minimal}: that is, when the complement of the arrangement has<br>torsion-free cohomology, then it is a minimal space. <br><br>We pass now to a more precise description of the contents of the<br>several parts of the thesis. <br><br>Chapters \ref{prerequisiti}, \ref{sottospazi} and<br>\ref{salvettisettepanella} are introductive, the original part can be<br>found at most in chapters \ref{sec:formula} and \ref{configuration}. <br><br>Chapter \ref{prerequisiti} is an introductory collection of the main<br>tools needed in the following parts. It includes: Orlik-Solomon algebra<br>and related topics, as the so called \emph{broken circuit bases}; the<br>definition of Salvetti complex; the main definitions and results of<br>the Discrete Morse Theory, following the original work by Forman<br>(\cite{forman,forman1}). <br><br><br>In chapter \ref{sottospazi} we deal with general<br>subspace arrangements. In section \ref{Gorformula} we recall<br>Goresky-MacPherson formula. We consider here the explicit example<br>given in \cite{jewell} of a subspace arrangement such that its<br>complement is not torsion-free. This arrangement is composed with six<br>codimensional-5 coordinate subspaces in $\R^{10}$ (we make complete<br>computation of the cohomology of the complement by using<br>Goresky-MacPherson formula).<br><br><br>In section \ref{spaziconfigurazione} we define generalized<br>$d-$configuration spaces $\mathcal{M}(\A)^{(d)}$,<br>and the generalized Salvetti complex <br>$\S^{(d)},$ whose cells correspond to all \emph{chains}<br>$(C\<F_1\<\dots\<F_d),$ where $C$ is a chamber and the $F_i$'s are<br>facets of the induced stratification $\Fi(\A)$ of $\R^n$ (and $\<$ is the<br>standard face-ordering in $\Fi(\A)$). <br><br>In chapter \ref{salvettisettepanella} we present the reduction of the<br>complex $\S=\S^{(1)}$ using discrete Morse theory, following \cite{salvsett}. <br>We define a system of polar coordinates in $\R^n$, and the induced <br>polar ordering on the stratification<br>$\Fi(\A).$ Next, <br>we define a gradient vector field $\Gamma$ on the set<br>of cells of $\S$; the critical cells of $\Gamma$ are in one-to-one<br>correspondence with the cells of a new $CW$-complex, which has the same<br>homotopy type as $\S.$ One can verify that the number of critical cells<br>of dimension $k$ equals the $k-$th Betti<br>number, so the latter $CW$-complex is minimal. <br><br><br>The main original part of our thesis is contained in the last two chapters.<br><br>In chapter \ref{sec:formula} we consider the two-dimensional case. For<br>any affine line arrangement $\A,$ we give explicit \emph{minimal} attaching<br>maps for the minimal two-complex corresponding to the polar gradient<br>vector field. After considering the central case, the proof is by<br>induction on the number of $0$-dimensional facets of $\A.$ <br><br>Of course, presentations of the fundamental<br>group of the complement follow straightforward from these explicit<br>boundary formulas. <br><br><br>In chapter \ref{configuration} we apply discrete Morse theory to the<br>complex $\S^{(d)}$. Even if the philosophy here is similar to that used<br>for $d=1$, the extension to the case $d>1$ is not trivial. To<br>construct a gradient field on $\S^{(d)},$ we have to consider on the<br>$i$th-component of the chains $(C\<F_1\<\dots\<F_d)\in \Sd$ <br>either the polar ordering which is<br>induced on the arrangement "centered" in the $(i+1)$th-component of<br>the chain, or the opposite of such ordering, according to the parity<br>of $d-i.$ Then we use a double induction over $d$ and the dimension of<br>a sub-arrangement of $\A.$ <br> <br><br>Several examples are considered in order to better illustrate our results.<br>

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