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A basis is presented for the RP" classes in y\n(BO). This basis is used to prove that every smooth involution (M, T) on a closed manifold bounds if its fixed point set is a disjoint union of odd-dimensional projective spaces of constant dimension.doi:10.1090/s0002-9939-1993-1136239-6 fatcat:ekyjhwra7nafdb4z64lm4dmjea