The generalized Lefschetz number and loop braid groups

TL;DR

This paper introduces loop braid groups as a three-dimensional extension of classical braid groups, linking algebraic representations to topological dynamics to analyze fixed and periodic points in 3D manifolds.

Contribution

It defines loop braid groups and connects their Burau matrix representations with the generalized Lefschetz number, extending classical 2D results to 3D dynamical systems.

Findings

Established a relationship between Burau matrices and the generalized Lefschetz number in 3D.

Provided estimates for the number of periodic points in 3D homeomorphisms.

Extended classical 2D theorems to three-dimensional topological dynamics.

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 for three-dimensional manifolds has been lacking. In this work, we introduce loop braid groups as a three-dimensional generalization of classical braid groups in order 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 with the generalized Lefschetz number. This result provides important information on the existence and interaction of fixed and periodic points. As an application of our theorem, we obtain an estimate for the number of periodic points. Our result extends a classical two-dimensional theorem to the three-dimensional setting, providing a framework in which the topological and algebraic aspects of loop braid groups can be used to study topological dynamical properties.