Smooth Structures on <i>M</i> × S <i>k</i>

Abstract This paper explores various differentiable structures on the product manifold $M \times \mathbb {S}^k$, 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 $M\times \mathbb {S}^k$ for $1\le k\le 10.$ These calculations enable us to further classify all manifolds that are homeomorphic to $\mathbb {C}P^2\times \mathbb {S}^k$, up to diffeomorphism, for each $4\le k\le 6$.