Survey on the Generalized R. L. Moore Problem

TL;DR

This survey reviews recent progress on the generalized R. L. Moore problem, focusing on characterizing codimension one manifold factors and the techniques used to solve specific cases.

Contribution

It provides an updated overview of results and methods related to the longstanding open problem in topology, emphasizing general position techniques.

Findings

Many special cases of the problem have been solved.

Efficient general position techniques are central to these solutions.

The survey highlights ongoing challenges and future directions.

Abstract

We give an updated extended survey of results related to the celebrated unsolved generalized R. L. Moore problem. In particular, we address the problem of characterizing codimension one manifold factors, i.e. spaces $X$ having the property that $X \times \mathbb{R}$ is a topological manifold. A main part of the paper is devoted to many efficient general position techniques, which have been used to solve special cases of this problem.