Coarse cohomology of configuration space and coarse embedding

We introduce a notion of equivariant coarse cohomology of the complement of a subspace in a metric space. We use this cohomology to define a notion of coarse cohomology of the configuration space of a metric space and develop tools to compute this cohomology under certain conditions. As an application of this theory, we show that certain classes in the coarse cohomology of configuration space obstruct coarse embedding between two metric spaces.