3:30 pm, Seminar Hall
Salvetti Complex Construction for Manifold Reflection Arrangements
Chennai Mathematical Institute.
The natural action of a reflection (Coxeter) group on a real vector space fixes a number of hyperplanes. The group acts fixed point freely on the complexified complement of the union of reflecting hyperplanes. The fundamental group of the complement is the pure Artin group and that of the quotient is the Artin group. The work of Salvetti provided cellular and simplicial complexes, constructed from the combinatorics of the hyperplane intersections, with the same homotopy type as the complexified complement and its quotient. This construction, popularly known as the Salvetti complex, has proved to be useful in deriving many algebraic properties of Artin groups.
In the present thesis we consider reflection groups that appear as groups of diffeomorphisms of smooth manifolds. Drawing parallels with the classical theory we define reflection arrangements on manifolds. The notion analogous to the complexified complement is the tangent bundle complement. The main result of the thesis is the construction of a simplicial complex following the Salvetti complex which has the same homotopy type as the tangent bundle complement. The principle ingredient we use is an equivariant form of the `nerve lemma', it allows us to construct simplicial complexes of a given homotopy type from a good cover of the space.
I will begin the talk by explaining the setting of manifold reflection arrangements. Then provide a sketch of the proof of the main theorem. Finally, I will discuss some explicit computations when the manifold is the unit sphere and the action is by isometries.