Embeddability of multiple cones

TL;DR

This paper proves that if the n-th cone over a Peano continuum embeds into Euclidean space of dimension n+2, then the original space embeds into the 2-sphere, solving a problem posed by Rosicki.

Contribution

It establishes a new embedding criterion linking cones over Peano continua to their embeddability into spheres, addressing a previously open problem.

Findings

n-th cone over X embeds into R^{n+2} implies X embeds into S^2

Solves a problem proposed by W. Rosicki

Provides new insights into embeddability of cones over continua

Abstract

The main result of this paper is that if $X$ is a Peano continuum such that its $n$-th cone $C^n(X)$ embeds into $\RR^{n+2}$ then $X$ embeds into $S^2$. This solves a problem proposed by W. Rosicki.