  We prove that, if n is a 2-power, the unordered configuration space C(RP^n,2)
cannot be immersed in R^{4n-2} nor embedded as a closed subspace of R^{4n-1},
optimal results, while if n is not a 2-power, C(RP^n,2) can be immersed in
R^{4n-3}. We also obtain cohomological lower bounds for the topological
complexity of C(RP^n,2), which are nearly optimal when n is a 2-power. We also
give a new description of the mod-2 cohomology algebra of the Grassmann
manifold G_{n+1,2}.
