The generalized Lefschetz number and loop braid groups

Abstract

We study the interplay between braid group theory and topological dynamics in three dimensions. While classical braid theory has been extensively applied to surface homeomorphisms to analyze fixed and periodic points, an analogous framework in three-dimensional manifolds has been lacking. In this work, we introduce the use of loop braid groups as a three-dimensional generalization of classical braid groups to investigate homeomorphisms of the 3-ball that leave invariant a finite collection of circles. In our main theorem we associate the Burau matrix representations of loop braid elements to the generalized Lefschetz number. This result provides important information on the existence and interaction of fixed and periodic points of such homeomorphisms. In addition, an application of our theorem gives an estimate of the number of their periodic points. Our theorem establishes a three-dimensional analogue of a classical result, providing the first framework that connects loop braid groups with Nielsen fixed point theory and topological dynamics in dimension three, providing a rich 3-dimensional framework, whose topological and algebraic aspects have been extensively investigated, for studying its topological dynamical properties.