Abstract: | We classify, up to diffeomorphism, all closed smooth manifolds homeomorphic to the complex projective n-space (mathbb {C}textbf {P}^{n}), where n=3 and 4. Let M2n be a closed smooth 2n-manifold homotopy equivalent to (mathbb {C}textbf {P}^{n}). We show that, up to diffeomorphism, M6 has a unique differentiable structure and M8 has at most two distinct differentiable structures. We also show that, up to concordance, there exist at least two distinct differentiable structures on a finite sheeted cover N2n of (mathbb {C}textbf {P}^{n}) for n=4,7 or 8 and six distinct differentiable structures on N10. |