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

TL;DR

This paper computes the inertia groups of highly connected 2n-manifolds, completing their diffeomorphism classification for all but one case, and reveals that homotopy and concordance inertia groups always vanish for these manifolds.

Contribution

It provides explicit calculations of inertia groups for (n-1)-connected 2n-manifolds, finishing a classification program initiated by Wall and applying advanced surgery techniques.

Findings

Inertia groups vanish for n ≠ 4,8,9.

Explicit inertia group computations for n=4,8,9.

Homotopy and concordance inertia groups always vanish.

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 In the cases $n=4,8$, our results recover unpublished work of Crowley--Nagy and Crowley--Olbermann.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.