Fundamental racks of braid spaces of complex reflection groups

Let $\Gamma$ 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 $G := \pi_1^{\operatorname{orb}}( \Gamma \backslash (X - \mathcal{H}))$ 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.