Coxeter transformation groups and reflection arrangements in smooth manifolds

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 which is equivariantly homotopy equivalent to the tangent bundle complement.