Last update: 3 May 2013

Notations and Conventions |

Summary |

Chapter I. Hyperplane Systems and Galleries |

§1. Pregalleries |

§2. Hyperplane systems |

§3. Tits galleries |

§4. Galleries |

§5. Simple hyperplane systems and signed galleries |

Chapter II. Artin groups |

§1. Coxeter and Artin groups |

§2. Linear Coxeter groups |

§3. The regular orbit space |

§4. Applications to abstract Artin groups |

Chapter III. Extended Artin groups |

§1. Generalised rootsystems and extended Coxeter groups |

§2. The regular orbit space and its fundamental group |

§3. The presentation of ${\pi}^{-1}\left(W\right)$ |

§4. Proof of the main theorem |

§5. Affine Artin groups |

Chapter IV. The $K(\pi ,1)\text{-problem}$ |

§1. Introduction |

§2. Stars and their hyperplane systems |

§3. An open cover of the universal covering space |

§4. The simplicial complex in terms of galleries |

§5. The Artin complex and the extended Artin complex |

§6. The contractibility problem |

§7. The direct pieces property; conjectures |

References |

- Homotopy equivalence curves is denoted by $\simeq \text{.}$
- The homotopy class of a curve $\gamma $ is denoted by $\left\{\gamma \right\}\text{.}$
- Let $a$ and $b$ be elements of a real or complex vectorspace. We denote by $[a,b]$ both the curve defined by $[0,1]\ni t\to (1-t)a+tb,$ as well as its image; the context should make it clear, which interpretation is meant. Also $[a,b)$ denotes $[a,b]-\left\{b\right\}\text{.}$
- "Connected" always means "arcwise connected".
- "Component" always means "arcwise connected component".
- The number of elements in a set $A$ is denoted by $\left|A\right|\text{.}$
- Let $\mathcal{C}$ be a collection of sets. Then $\text{nerf}\left(\mathcal{C}\right)$ denotes the abstract simplicial complex with $\mathcal{C}$ as vertices and a finite subcollection forms a simplex iff the intersection is nonempty.
- The positive integers, the nonnegative integers, the integers, the real numbers, the complex numbers and the nonzero complex numbers are denoted by respectively: $\mathbb{N},$ ${\mathbb{Z}}^{+},$ $\mathbb{Z},$ $\mathbb{R},$ $\u2102,$ ${\u2102}^{*}\text{.}$
- A finite dimensional real vector space is often denoted by $\mathbb{V},$ $\mathbb{V}\prime ,$ etc.
- Definitions or recalls of definitions are indicated by $:=$ or $=:,$ where the lefthandside (resp. righthandside) is being defined.
- References in the same chapter are given as (3.2) e.g. References to another chapter as (II.3.2) e.g.

We study the homotopy type of a space $Y$ obtained by deleting a union of complex hyperplanes from a convex complex domain $\Omega \text{.}$ In particular we investigate the fundamental group (or -groupoid) and the question under which circumstances the higher homotopy groups vanish. In this case $Y$ is called a $K(\pi ,1)\text{.}$ For instance, if $\Omega =\u2102,$ then this is for $Y:=\Omega -\left\{0\right\}={\u2102}^{*}$ the case, since by the exponential map $\u2102\to {\u2102}^{*},$ it turns out that the contractible space $\u2102$ is the universal covering space of ${\u2102}^{*}\text{.}$

Sometimes the space $Y$ is just the set of points with a trivial stabiliser of a group $W,$ which acts
on $\Omega $ and $X:=Y/W$
is then the so-called *regular orbit space*. The latter occurs often as the complement of the semiuniversal deformation of a singularity, which has
$W$ as its monodromy group. For instance this is the case for simple singularities (Grothendieck, Brieskorn [Bri1970] 1970) and also for
unimodular singularities (Pinkham, Looijenga [Loo1981] 1976). The question of the homotopy type of this complement (in particular the fundamental group) and the
conjecture of Arnold, Pham and Thorn, that this space is a $K(\pi ,1),$
has been the most important motivation for this study. The conjecture is proved for simple singularities by Deligne [BSa1972] (1972). This thesis is highly inspired by his proof.

The domain $\Omega $ is in ${\mathbb{V}}_{\u2102},$
the complexificatlon of a finite dimensional real vector space $\mathbb{V},$ and is always of the form
${\text{Im}}^{-1}\left(H\right),$ where
$H$ is an open convex set (often a cone) in $\mathbb{V}\text{.}$ In the socalled *simple case*
we delete from $\Omega $ a union of complex hyperplanes, which are complexifications of real hyperplanes from a
collection $\mathcal{M},$ which is locally finite on $H\text{.}$ To treat the situation
of *generalised root systems* ([Loo1980]) (for the unimodular singularities), we give a slightly more general definition of a *hyperplane system*, which
allows that also "complex affine" hyperplanes are deleted.

Now we discuss briefly the contents of each chapter.

In chapter I we make an elementary topological observation, which leads to a description of the fundamental groupoid of a space, with a given open cover consisting of
simply connected open sets. The calculations of presentations of fundamental groups in this thesis are all based on this observation. In particular we do not use
theorems of Van Kampen, Zariski or Lefschetz. "We give a combinatorial description of the fundamental groupoid of the space $Y\text{.}$
The notion of a *gallery* is the most important tool. The meaning of this notion here is somewhat more general then the meaning in the work of Tits
(and Deligne), who introduced this beautiful terminology.

In chapter II we apply these results in the situation of the so-called *linear Coxeter groups* of Vinberg
[Vin1971]. We calculate a presentation of the fundamental group of the regular orbit space, which gives a geometrical interpretation of all the so-called
*Artin groups* [BSa1972]. This generalises a result of Brieskorn [Bri1971], who had proved this already for finite groups and of Nguyen Viet Dung
[Ngu1983], who treated the affine Coxeter groups. At the same time the theorem is placed in its natural context, namely the theory of Vinberg. We apply this
interpretation by proving some results about abstract Artin groups. These results are also obtained by Appel and Schupp [ASc1983]. However, by their methods
(small cancellation), they have to assume that the Artin groups are of *large type*. We do not need this pretty restrictive condition.

In chapter III we consider the situation of a so-called *extended Coxeter group* $W\text{.}$ which arises from a
generalised root system. Now we use the complete power of the results of chapter I to derive a presentation of the fundamental group of the regular orbit space.
It is nice to see how one can obtain this presentation (analogous as in the case of Artin groups) by changing the obvious presentation of the group
$\stackrel{\sim}{W}$ in a suitable way and then deleting some relations. We propose to call this group an *extended Artin group*,
although it is not an extension in the usual sense. If the group $\stackrel{\sim}{W}$ is of finite type, then
$\stackrel{\sim}{W}$ is itself again a Coxeter group. The Artin group associated to this (affine) Coxeter group is isomorphic to
the extended Artin group associated to $\stackrel{\sim}{W}$ and we establish an explicit isomorphism.

In chapter IV we discuss the $K(\pi ,1)\text{-problem.}$ Suppose
we have a simple hyperplane system, which is *central* i.e. $H=\mathbb{V}$ and the intersection of all (real) hyperplanes
consists of the origin only (in particular nonempty). Then the components of $\mathbb{V}=\underset{M\in \mathcal{M}}{\cup}\mathcal{M}$
are cones. If these *chambers* are simplicial cones then $Y$ is a $K(\pi ,1)$
as is proved by Deligne [Del1972] in 1972. His proof consists in fact of two steps: First he constructs a simplicial complex, which has the homotopy type of the
universal covering space of $Y\text{.}$ Then he proves that this complex is contractible. The discovery that for the first
step the central condition nor the simplicial condition are necessary, has encouraged us to try to generalise the result such that it was applicable for instance on
the regular orbit space of infinite Coxeter groups (some results are obtained in this direction, for instance Okonek [Oko1979]).

Unfortunately this goal is not completely achieved. The first step is done in great generality. It leads to a simplicial complex, which has the homotopy type of
the universal covering space of $Y$ in the case that the hyperplane system is already
$\text{locally-}K(\pi ,1)\text{.}$
We give detailed descriptions of this complex. First in general in terms of galleries. Then also in terms of Artin groups and extended Artin groups, if the system
arised from a linear Coxeter group (respectively a generalised root system). This leads to the notions of *Artin complex* and *extended Artin complex*.
These constructions are similar to the well known *Coxeter complex* of a Coxeter group.

The "proof" of the second step uses a still unproven conjecture. Yet these partial results are interesting. There are indications that the problem is of the same
difficulty as the so-called *word problem* for Artin groups. For Artin groups of finite type this was solved by Brieskorn and Saito [BSa1972] and Deligne
[Del1972]. For Artin groups of large type Appel and Schupp [ASc1983] found a solution with the help of small cancellation theory.

This is an excerpt of a thesis entitled *The Homotopy Type of Complex Hyperplane Complements*, written by Harm van der Lek in 1983.