Inertia groups in the metastable range

Abstract

We prove that the inertia groups of all sufficiently-connected, high-dimensional $(2n)$-manifolds are trivial. This is a key step toward a general classification of manifolds in the metastable range. Specifically, for $m\gg 0$ and $k>5/12$, suppose $M$ is a $\lfloor km\rfloor$-connected, smooth, closed, oriented $m$-manifold and $\Sigma$ is an exotic $m$-sphere. We prove that, if $M♯\Sigma$ is diffeomorphic to $M$, then $\Sigma$ bounds a parallelizable manifold. Our proof is built on an understanding of the second extended power functor in Pstrągowski's category of synthetic spectra.

arXiv commentv2: Minor corrections and improvements. 22 pages. Comments still welcome!