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The generalized Lefschetz number and loop braid groups

Stavroula Makri

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.

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 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.
Paper Structure (13 sections, 6 theorems, 49 equations, 4 figures)

This paper contains 13 sections, 6 theorems, 49 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Generalized Lefschetz number
  3. loop braid groups
  4. Main result
  5. Twisted representation of $\mathrm{LB}_n$
  6. Compactification
  7. Linking number
  8. Abelianized generalized Lefschetz number
  9. Action on the universal covering space
  10. Main theorem
  11. Examples and applications
  12. Example
  13. Application

Key Result

Theorem 1

Let $b = b_{f,C}$. Then where $S(b)=\bar{R}(b)-R(b)$.

Figures (4)

  • Figure 1: The generators $\sigma_i$ and $\rho_i$.
  • Figure 2: Universal covering space of $X$ for $n=2$.
  • Figure 3: On the left hand side we see the $2$-cells $b_i$ and $b_{i+1}$ and on the right hand side we see the image of $b_i$ under the action of $\sigma_i$.
  • Figure 4: The image of $b_{i+1}$ under the action of $\sigma_i$.

Theorems & Definitions (18)

  • Theorem 1
  • Theorem 2
  • Definition 2.1
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Theorem 3.4: Brendle--Hatcher, brendle2013configuration
  • Remark 3.5
  • Remark 3.6
  • Definition 3.7
  • ...and 8 more