Inertia groups and smooth structures of $(n-1)$-connected $2n$-manifolds

Let $M^{2n}$ denote a closed $(n-1)$-connected smoothable topological $2n$-manifold. We show that the group $\mathcal{C}(M^{2n})$ of concordance classes of smoothings of $M^{2n}$ is isomorphic to the group of smooth homotopy spheres ${\bar{\mathrm{\Theta }}}_{2n}$ for $n=4$ or $5$, the concordance inertia group $I_c(M^{2n})=0$ for $n=3$, $4$, $5$ or $11$ and the homotopy inertia group $I_h(M^{2n})=0$ for $n=4$. On the way, following Wall’s approach \cite{Wal67} we present a new proof of the main result in \cite{KS07}, namely, for $n=4$, $8$ and $H^n(M^{2n};\mathbb{Z})\cong \mathbb{Z}$, the inertia group $I(M^{2n})\cong {\mathbb{Z}}_2$. We also show that, up to orientation-preserving diffeomorphism, $M^8$ has at most two distinct smooth structures; $M^{10}$ has exactly six distinct smooth structures and then show that if $M^{14}$ is a $\pi$-manifold, $M^{14}$ has exactly two distinct smooth structures.