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

Ramsey, Patrick and Evans, Jonny (2025) Thick braids and other non-trivial homotopy in configuration spaces of hard discs. PhD thesis, Lancaster University.

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Abstract

We study ordered configuration spaces of hard discs inside a unit disc, focusing in some cases on small numbers of discs, but generalising where possible. These are a natural generalisation of ordered configuration spaces of points inside the unit disc. As the radius of the hard discs increases, the homotopy-type is known to change at finitely many ‘critical radii’, and we classify the critical radii for between one and five discs. We study the configuration spaces just above the least critical radius through their homotopy groups (which are generated by non-contractible maps from the sphere into the configuration space). By observing that the disc centres cannot all lie on one line in the unit disc, we find a non-contractible S^{n−3} in the configuration space of n discs just beyond the first critical radius, which vanishes below this radius. In the case n = 4, we find a circle action on the configuration space beyond the first critical radius, and construct the quotient space. By retracting the quotient space onto a graph, we show that the configuration space is homotopy-equivalent to the product of a circle with a graph of Euler characteristic 11. We explore the persistence of the non-contractible S^{n−3} beyond the first critical radius as the unit disc is deformed into an ellipse, and demonstrate that there remains a non-contractible map from S^{n−3} into the configuration space of n hard discs inside an ellipse of any eccentricity. We consider the homotopy groups in dimension less than n − 3 of configuration spaces of hard discs inside a unit disc beyond the first critical radius. We conjecture that these are isomorphic to the homotopy groups in the same dimension of the configuration space of points. We suggest a method of proof by defining a flow on the configuration space with certain properties, and demonstrate such a flow on the configuration space of three points.