A simplification problem in manifold theory

Two smooth manifolds M and N are called R-diffeomorphic if their product with the real line are diffeomorphic. We consider the following simplification problem: does 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 classification results in dimensions 3 and 4. This paper presents what is currently known about the subject as well as some new results about classifications of R-diffeomorphisms.