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We prove that, if n is a 2-power, the unordered configuration space $$C(RP^n,2)$$ cannot be immersed in $${\mathbb {R}}^{4n-2}$$ nor embedded as a closed subspace of $${\mathbb {R}}^{4n-1}$$, optimal results, while if n is not a 2-power, $$C(RP^n,2)$$ can be immersed in $${\mathbb {R}}^{4n-3}$$. 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 $$G_{{n+1},2}$$.

On the unordered configuration space $$C(RP^n,2)$$