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

TL;DR

This paper investigates the smooth structures and inertia groups of certain highly connected 2n-manifolds, establishing isomorphisms with homotopy spheres and classifying the number of smooth structures in specific dimensions.

Contribution

It provides new proofs and results on the inertia and concordance inertia groups of (n-1)-connected 2n-manifolds, including classifications of smooth structures in dimensions 8, 10, and 14.

Findings

(M^{2n}) e2 a0 ext{is isomorphic to} \u00a0 ar{\u03b8}_{2n} for n=4,5

I_c(M^{2n})=0 for n=3,4,5,11

I_h(M^{2n})=0 for n=4

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 $\overline{\Theta}_{2n}$ 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.