Salvetti Complex Construction for Manifold Reflection Arrangements

Artin groups are a natural generalization of braid groups and are closely related to Coxeter groups in the following sense. There is a faithful representation of a Coxeter group W as a linear reflection group on a real vector space. The group acts properly and fixes a union of hyperplanes. The W-action extends as a covering space action to the complexified complement of these hyperplanes. The fundamental groups of the complement and that of the orbit space are respectively the pure Artin group and the Artin group. For finite Coxeter groups Deligne proved that the associated complement is an aspherical space. Later, 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. These cell complexes are important because a lot of algebraic properties of Artin groups were discovered using their combinatorial and topological aspects. In this thesis we consider Coxeter groups that appear as groups of diffeomorphisms. To be precise, given a smooth manifold, a reflection is an order-2 auto-diffeomorphism that locally behaves like usual Euclidean reflections. Under suitable topological conditions the discrete subgroup generated by finitely many such ‘manifold reflections’ is a Coxeter group. We use this group data to introduce the notion of reflection arrangements on manifolds. These arrangements are a collection of finitely many codimension-1 submanifolds such that locally they resemble a hyperplane arrangement and provide a stratification of the manifold with combinatorial properties analogous to those in the classical case. Since the action of such a group is smooth it extends naturally to the tangent bundle; this observation leads to the definition of a tangent bundle complement as a generalization of the complexified complement. The main aim of this thesis is to demonstrate how the combinatorial data of the reflection group can be used to build a regular cell complex with the same homotopy type as that of the tangent bundle complement. We show that this homotopy equivalence respects the reflection group action hence we also obtain a combinatorial model for the quotient space. These results lead to a manifold theoretic generalization of Artin groups. The proof of our main theorem uses an equivariant version of the nerve lemma. We use the incidence relations among the strata of the manifold to construct an open cover of the tangent bundle complement that is invariant under the group action. We further show that this open cover satisfies the hypotheses of the nerve lemma. Our work is a generalization of the seminal result of Salvetti and to best of our knowledge is new.