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

Abstract

In this paper, we compute the inertia groups of $(n-1)$-connected, smooth, closed, oriented $2n$-manifolds where $n \geq 3$. As a consequence, we complete the diffeomorphism classification of such manifolds, finishing a program initiated by Wall sixty years ago, with the exception of the $126$-dimensional case of the Kervaire invariant one problem. In particular, we find that the inertia group always vanishes for $n \neq 4,8,9$ -- for $n \gg 0$, this was known by the work of several previous authors, including Wall, Stolz, and Burklund and Hahn with the first named author. When $n = 4,8,9$, we apply Kreck's modified surgery and a special case of Crowley's $Q$-form conjecture, proven by Nagy, to compute the inertia groups of these manifolds. In the cases $n=4,8$, our results recover unpublished work of Crowley--Nagy and Crowley--Olbermann. In contrast, we show that the homotopy and concordance inertia groups of $(n-1)$-connected, smooth, closed, oriented $2n$-manifolds with $n \geq 3$ always vanish.