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

Abstract

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 $\bar{\Theta}_{2n}$ 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 [16] we present a new proof of the main result in [9], 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.