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We compute the integral homology and cohomology groups of configurationspaces of two distinct points on a given real projective space. The explicitanswer is related to the (known multiplicative structure in the) integralcohomology---with simple and twisted coefficients---of the dihedral group oforder 8 (in the case of unordered configurations) and the elementary abelian2-group of rank 2 (in the case of ordered configurations). As an application,we complete the computation of the symmetric topological complexity of realprojective spaces of dimension 2^i + d for d=0,1,2.

The integral cohomology groups of configuration spaces of pairs of points in real projective spaces