A simplification problem in manifold theory

Abstract

Two smooth manifolds $M$ and $N$ are called $\mathbb{R}$-diffeomorphic if $M\times \mathbb{R}$ is diffeomorphic to $N\times \mathbb{R}$. We consider the following simplication problem: does $\mathbb{R}$-diffeomorphism imply diffeomorphism or homeomorphism? For compact manifolds, analysis of this problem relies on some of the main achievements of the theory of manifolds, in particular the h- and s-cobordism theorems in high dimensions and the spectacular more recent classication results in dimensions 3 and 4. This paper presents what is currently known about the subject as well as some new results about classications of $\mathbb{R}$-diffeomorphisms.