Decompositions of $\mathbb{R}^n, n \geq 4,$ into convex sets generate codimension 1 manifold factors

TL;DR

This paper proves that decomposing higher-dimensional Euclidean spaces into convex sets results in quotient spaces that are manifold factors of codimension one, with specific topological properties like the disjoint arc-disk property.

Contribution

It establishes that upper semicontinuous decompositions of 4-dimensional Euclidean space into convex sets produce quotient spaces that are codimension one manifold factors, advancing understanding of such decompositions.

Findings

4-dimensional Euclidean space quotients are manifold factors

Quotients have the disjoint arc-disk property

Decompositions into convex sets preserve manifold structure

Abstract

We show that if $G$ is an upper semicontinuous decomposition of $\mathbb{R}^n$, $n \geq 4$, into convex sets, then the quotient space $\mathbb{R}^n/G$ is a codimension one manifold factor. In particular, we show that $\mathbb{R}^n/G$ has the disjoint arc-disk property.