Inertia groups in the metastable range

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 $Σ$ is an exotic $m$-sphere. We prove that, if $M \sharp Σ$ is diffeomorphic to $M$, then $Σ$ bounds a parallelizable manifold. Our proof is built on an understanding of the second extended power functor in Pstrągowski's category of synthetic spectra.

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