Abstract This paper explores various differentiable structures on the product manifold M Sᵏ, where M is either a 4-dimensional closed, oriented, smooth manifold or a simply connected 5-dimensional closed, smooth manifold. We identify the possible stable homotopy types of M and use it to calculate the concordance inertia group and the concordance structure set of M Sᵏ for 1 k 10. These calculations enable us to further classify all manifolds that are homeomorphic to CP² Sᵏ, up to diffeomorphism, for each 4 k 6.
Basu et al. (Tue,) studied this question.