# Linking topological spheres

## TL;DR

This paper constructs a specific embedding of a circle in five-dimensional space demonstrating that no 3-sphere can be topologically linked with it, revealing new insights into high-dimensional linking phenomena.

## Contribution

It provides an explicit example of a topological embedding in -dimensional space where the third homotopy group of the complement vanishes, showing no linking with 3-spheres.

## Key findings

- Existence of a circle embedding in -space with trivial -homotopy of the complement
- No 3-sphere can be linked with this embedding
- Advances understanding of high-dimensional linking and embedding properties

## Abstract

There is a topological embedding $\iota:\mathbb{S}^1\to\mathbb{R}^5$ such that $\pi_3(\mathbb{R}^5\setminus\iota(\mathbb{S}^1))=0$. Therefore, no $3$-sphere can be linked with $\iota(\mathbb{S}^1)$.