Coxeter transformation groups and reflection arrangements in smooth manifolds

TL;DR

This paper extends the theory of Artin and Coxeter groups by representing Coxeter groups as diffeomorphism groups of smooth manifolds, constructing associated cell complexes, and proving homotopy equivalences in this geometric setting.

Contribution

It introduces Coxeter transformation groups acting on smooth manifolds and generalizes Salvetti's cell complex construction to this new context.

Findings

Coxeter groups can be realized as subgroups of diffeomorphisms of smooth manifolds.

A cell complex homotopy equivalent to the tangent bundle complement is constructed.

Salvetti's theorems are proved within this geometric framework.

Abstract

Artin groups are a natural generalization of braid groups and are well-understood in certain cases. Artin groups are closely related to Coxeter groups. There is a faithful representation of a Coxeter group $W$ as a linear reflection group on a real vector space $V$. The group acts properly and fixes a union of hyperplanes. The $W$-action extends as the covering space action to the complexified complement of these hyperplanes. The fundamental groups of the complement and the orbit space are the pure Artin group and the Artin group respectively. For the Artin groups of finite type Deligne proved that the associated complement is aspherical. Using the Coxeter group data Salvetti gave a construction of a cell complex which is a $W$-equivariant deformation retract of the complement. This construction was independently generalized by Charney and Davis to the Artin groups of infinite type. A lot of algebraic properties of these groups were discovered using combinatorial and topological properties of this cell complex. In this paper we represent a Coxeter group as a subgroup of diffeomorphisms of a smooth manifold. These so-called Coxeter transformation groups fix a union of codimension-$1$ (reflecting) submanifolds and permute the connected components of the complement. Their action naturally extends to the tangent bundle of the ambient manifold and fixes the union of tangent bundles of these reflecting submanifolds. Fundamental group of the tangent bundle complement and that of its quotient serve as the analogue of pure Artin group and Artin group respectively. The main aim of this paper is to prove Salvetti's theorems in this context. We show that the combinatorial data of the Coxeter transformation group can be used to construct a cell complex homotopy equivalent to the tangent bundle complement and that this homotopy equivalence is equivariant.