Analyzing the Kuperberg Plug

Abstract

In 1950, the mathematician Herbert Seifert posed a question in the study of dynamical systems: Does every non-vanishing vector field on the 3-sphere S3 admit a periodic orbit?

This question, now known as the Seifert Conjecture, sits at the intersection of topology and dynamics, touching on fundamental issues about the structure of flows on manifolds.

The 3-sphere S3 is particularly significant due to the fact that its dimensionality is low enough that we cannot easily introduce trivial ways to eliminate orbits. Over the following decades, the conjecture inspired a group of mathematicians to develop tools to investigate flows on S3. Eventually leading to the construction of the Kuperburg plug, an a-periodic smooth non-vanishing manifold.