A simplification problem in manifold theory

TL;DR

This paper explores whether R-diffeomorphism implies diffeomorphism or homeomorphism for manifolds, reviewing known results and presenting new classifications of R-diffeomorphisms, especially in low and high dimensions.

Contribution

It provides a comprehensive overview of the simplification problem in manifold theory and introduces new classification results for R-diffeomorphisms.

Findings

Analysis of R-diffeomorphism implications for manifold classification

New results on R-diffeomorphism classifications in various dimensions

Review of key theorems like h- and s-cobordism in the context of the problem

Abstract

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.