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An exact sequence for generalized string links over surfaces

Juliana Roberta Theodoro de Lima

TL;DR

This work extends Goldberg's exact-sequence framework for link-homotopy braids from the disk to generalized string links on closed orientable surfaces of genus $g\ge 1$. It defines $\widehat{PB}_n(M)$ as the quotient of the pure surface braid group by link-homotopically trivial braids, introduces the injective map $\hat{f}_n$ from the disk case, and the surjective map $\hat{\theta}_n$ to $\pi_1(M)^n$, proving $\ker(\hat{\theta}_n) = \langle \mathrm{Im} \hat{f}_n \rangle^N$. Consequently, it yields a nonabelian short exact sequence linking $\widehat{PB}_n(\mathbb{D})$, $\widehat{PB}_n(M)$, and $\pi_1(M)^n$, providing a structural tool for orderability questions and a bridge between disk and surface braid theories. This advances the algebraic understanding of generalized string links on surfaces and establishes a framework for further investigations of their link-homotopy properties.

Abstract

In this work we extend Goldberg result \cite{Goldberg} for generalized string links over closed, connected and orientable surfaces of genus $g \geq 1$, i.e., different from the sphere (up to link-homotopy).

An exact sequence for generalized string links over surfaces

TL;DR

This work extends Goldberg's exact-sequence framework for link-homotopy braids from the disk to generalized string links on closed orientable surfaces of genus . It defines as the quotient of the pure surface braid group by link-homotopically trivial braids, introduces the injective map from the disk case, and the surjective map to , proving . Consequently, it yields a nonabelian short exact sequence linking , , and , providing a structural tool for orderability questions and a bridge between disk and surface braid theories. This advances the algebraic understanding of generalized string links on surfaces and establishes a framework for further investigations of their link-homotopy properties.

Abstract

In this work we extend Goldberg result \cite{Goldberg} for generalized string links over closed, connected and orientable surfaces of genus , i.e., different from the sphere (up to link-homotopy).
Paper Structure (7 sections, 14 theorems, 16 equations, 9 figures)

This paper contains 7 sections, 14 theorems, 16 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Homotopy Generalized String Links
  3. Braids over Surfaces
  4. Generalized String Links over Surfaces
  5. An Exact Sequence for Link-Homotopy Braid Groups
  6. Main Theorem
  7. The well definition of $\hat{f}_{n}$ and proof of Theorem \ref{['Teorema3.3.1']}

Key Result

Theorem 2.2

Gonzales If $M$ is a closed, connected and orientable surface of genus $g \geq 1$ (different from the sphere), then $PB_{n}(M)$ admits the following presentation: (PR1)$a_{n, 1}^{-1}a_{n, 2}^{-1}\cdots a_{n, 2g}^{-1}a_{n, 1}a_{n, 2}\cdots a_{n, 2g}= \prod_{i= 1}^{n-1}{T_{i, n-1}^{-1}T_{i, n}}$; (PR2)$a_{i, r}A_{j, s}= A_{j, s}a_{i, r}, \space 1\leq i<j \leq n; 1 \leq r, s \leq 2g; r\neq s$; (PR3)

Figures (9)

  • Figure 1: A braid "through the wall" $\beta$ over the $2$-dimensional torus.
  • Figure 2: Two isotopic braids.
  • Figure 3: Generators of $PB_{n}(M)$.
  • Figure 4: Generators of $B_{n}(M)$.
  • Figure 5: A generalized string link $\sigma$ over the $2$-dimensional torus.
  • ...and 4 more figures

Theorems & Definitions (22)

  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Proposition 2.7
  • Proposition 2.8
  • Theorem 2.9
  • Proposition 2.10
  • ...and 12 more