The Whitehead group and stably trivial $G$-smoothings

TL;DR

This paper explores the differences between classical and equivariant smooth structures on manifolds, demonstrating that infinitely many $G$-smoothings exist for certain $G$-manifolds and analyzing their properties.

Contribution

It introduces controlled $h$-cobordisms to construct infinitely many $G$-smoothings, contrasting with the finite smooth structures in the non-equivariant case.

Findings

Constructs infinitely many $G$-smoothings of a $G$-manifold.

Shows $G$-smoothings are isotopic after product with $ eal$.

Highlights differences between classical and equivariant smooth structures.

Abstract

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 $\mathbb{R}$.