Fundamental racks of braid spaces of complex reflection groups

Abstract

Let $Γ$ be a complex reflection group acting on the complex affine or hyperbolic space $X$ with the set of reflecting hyperplanes $\mathcal{H}$. We define an augmented rack $(G, \mathcal{K}, p)$ associated to the orbifold fundamental group $G := π_1^{\operatorname{orb}}( Γ\backslash (X - \mathcal{H}))$ which plays the role of the fundamental rack of a framed link complement as defined by Fenn and Rourke. This yields representations of the orbifold fundamental group $G$ on the cohomology of the associated rack space.