The Whitehead group and stably trivial $G$-smoothings

A closed manifold $M$ of dimension at least $5$ has only finitely many smooth structures. Moreover, smooth structures of $M$ are in bijection with smooth structures of $M\times \mathbb{R}$. Both of these statements are false equivariantly. In this paper, we use controlled $h$-cobordisms to construct infinitely many $G$-smoothings of a $G$-manifold $X$. Moreover, these $G$-smoothings are isotopic after taking a product with $\mathbb{R}$.