Smooth Structures on the product of 3-connected 8-manifolds with spheres

Let $M$ be a closed, 3-connected, 8-dimensional smooth manifold. In this paper, we compute the concordance inertia group of the product manifold $M\times {\mathbb{S}}^k$ for $1\le k\le 14$ and classify all smooth manifolds homeomorphic to $M\times {\mathbb{S}}^k,$ up to concordance for $1\le k\le 10.$ Moreover, we provide a diffeomorphism classification of smooth manifolds homeomorphic to $M\times {\mathbb{S}}^1,$ where $H^4(M;\mathbb{Z})=\mathbb{Z}\mathrm{.}$