Thick braids and other non-trivial homotopy in configuration spaces of hard discs

Patrick Ramsey

Published 2023-11-28Version 1

We study the homotopy groups of the configuration space of discs inside a unit disc just beyond the first critical radius. We find a non-trivial `thick braid' (loop in the configuration space), which is trivial in the ordered configuration space of points when replacing each disc by its centre, in the case $n=4$. We find a non-contractible $(n-3)$-sphere for $n\ge 5$, where $r$ is between the first two critical radii, and explore the persistence of this homotopy class when the ambient unit disc is deformed. For sufficiently large $n$, we demonstrate the existence of non-trivial $\pi_k$-classes with $k<n-3$ beyond higher critical radii.