Unit sphere fibrations in Euclidean space

TL;DR

This paper investigates conditions under which open sets in Euclidean space can be fibered by unit spheres, establishing dimension bounds, linking properties, and constructing explicit fibrations for specific dimensions.

Contribution

It proves a dimension bound for sphere fibrations in Euclidean space and characterizes linking and existence conditions for certain sphere dimensions.

Findings

Dimension bound: d ≥ 2n+1 for sphere fibrations

Linked spheres occur when d=2n+1

Explicit fibrations constructed for n in {0, 1, 3, 7}

Abstract

We show that if an open set in $\mathbb{R}^d$ can be fibered by unit $n$-spheres, then $d \geq 2n+1$, and if $d = 2n+1$, then the spheres must be pairwise linked, and $n \in \left\{ 0, 1, 3, 7 \right\}$. For these values of $n$, we construct unit $n$-sphere fibrations in $\mathbb{R}^{2n+1}$.