On the action of $\Theta ^{n}$. I

Abstract:

We prove two theorems about the inertia groups of closed, smooth, simply-connected $n$-manifolds. Theorem A shows that, in certain dimensions, the special inertia group, unlike the full inertia group, can never be equal to ${\Theta ^n}$; Theorem B shows, in $\operatorname {dimensions} \equiv 3\bmod 4$, how to construct explicit closed $n$-manifolds ${M^n}$ such that $\Theta (\partial \pi )$ is contained in the inertia group of ${M^n}$.