Fundamental racks of braid spaces of complex reflection groups

Let $\mathrm{\Gamma }$ 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 $G:={\pi }_1^{\mathrm{orb}}(\mathrm{\Gamma }\mathrm{\backslash }(X-\mathcal{H}))$ 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.