Abstract In this paper, we compute the inertia groups of (n − 1) (n-1) -connected, smooth, closed, oriented 2 n 2n -manifolds, where n ≥ 3 n 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 ≠ 4, 8, 9 n 4, 8, 9 ; for n ≫ 0 n 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 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 In the cases n = 4, 8 n=4, 8, our results recover unpublished work of Crowley–Nagy and Crowley–Olbermann.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) (n-1) -connected, smooth, closed, oriented 2 n 2n -manifolds with n ≥ 3 n 3 always vanish.
Senger et al. (Wed,) studied this question.