Enumerating Smooth Structures on $\mathbb{C}P^3\times\mathbb{S}^k$

In this paper, we compute the concordance inertia group of the product $M \times \mathbb{S}^k$, where $M$ is a simply connected, closed, smooth 6-manifold, for $1 \leq k \leq 10$, using known low-dimensional computations of the stable homotopy groups of spheres. Specifically, for $M = \mathbb{C}P^3$, we determine the inertia group of $\mathbb{C}P^3 \times \mathbb{S}^k$ for $2 \leq k \leq 7, k \neq 6$, and establish a diffeomorphism classification of all smooth manifolds homeomorphic to $\mathbb{C}P^3 \times \mathbb{S}^k$ for $1 \leq k \leq 7$.