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

Abstract

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 $1\le k\le 10$, 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 $2\le k\le 7,k\ne 6$, and establish a diffeomorphism classification of all smooth manifolds homeomorphic to $\mathbb{C}{P}^3\times {\mathbb{S}}^k$ for $1\le k\le 7$.

34 pages. The Abstract and title have been changed. Comments are welcome