The cohomology ring away from 2 of configuration spaces on real projective spaces

Let R be a commutative ring containing 1/2. We compute the R-cohomology ring of the configuration space F(m,k) of k ordered points in the m-dimensional real projective space. The method uses the observation that the orbit configuration space of k ordered points in the m-dimensional sphere (with respect to the antipodal action) is a 2^k-fold covering of F(m,k). This implies that, for odd m, the Leray spectral sequence for the inclusion of F(m,k) in the k-fold Cartesian self power of the m-dimensional real projective space collapses after its first non-trivial differential, just as it does when the projective space is replaced by a complex projective variety. The method also allows us to handle the R-cohomology ring of the configuration space of k ordered points in a punctured real projective space. Lastly, we compute the Lusternik-Schnirelmann category and all of the higher topological complexities of some of the auxiliary orbit configuration spaces.