Inertia groups in the metastable range

TL;DR

This paper proves that the inertia groups of highly-connected, high-dimensional manifolds are trivial, advancing the classification of manifolds in the metastable range by linking exotic spheres to bounding parallelizable manifolds.

Contribution

It establishes the triviality of inertia groups for sufficiently-connected high-dimensional manifolds, a key step in classifying manifolds in the metastable range.

Findings

Inertia groups of certain high-dimensional manifolds are trivial.

Exotic spheres that preserve the manifold's diffeomorphism class bound parallelizable manifolds.

The proof utilizes the second extended power functor in synthetic spectra.

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 $M \sharp \Sigma$ is diffeomorphic to $M$, then $\Sigma$ bounds a parallelizable manifold. Our proof is built on an understanding of the second extended power functor in Pstr\k{a}gowski's category of synthetic spectra.