Smooth Structures on M × S k

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.