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Action of automorphisms of pure braid groups on homotopy groups of two-sphere

Ilya Alekseev, Vasily Ionin, Mikhail Mikhailov

TL;DR

This work analyzes how automorphisms of the pure braid group ${P_n}$ act on the homotopy groups $\pi_n(S^2)$ through the Moore complex of a Delta-group built from Brunnian braids. It proves that the subgroups $Z_n=\mathrm{Brun}_n\cap\ker(\partial_n)$ and $Bd_n=\partial_{n+1}(\mathrm{Brun}_{n+1})$ are characteristic in $P_n$, inducing a well-defined action on $\pi_n(S^2)$, and provides explicit computations for small $n$ (notably $r_2$ is an isomorphism and $r_3$ has image $\{\mathrm{id},-\mathrm{id}\}$). The paper also gives detailed expressions for the relevant subgroups (Brun$_n$ and Bd$_n$) and proves invariance under the key automorphism classes ${\rm Aut}_c(P_n)$, $w_n$, and ${\rm Aut}(B_n)$. It culminates with a concrete demonstration at $n=3$ that the induced action is nontrivial and discusses a general conjecture that ${\rm Im}(r_n)=\{\mathrm{id},-\mathrm{id}\}$ for all $n$, along with corrections to earlier Delta-group arguments in the literature. These results connect Brunnian braids with the homotopy-theoretic structure of $S^2$ and illuminate how braid automorphisms modulate higher homotopy data.

Abstract

We examine the Moore complex of the Delta-group structure related to the pure braid groups and introduced by Berrick, Cohen, Wong, and Wu. We prove that the cycle and the boundary groups are invariant under all automorphisms of the pure braid groups, and thereby, we extend the results of Li and Wu on the reflection automorphism. We conclude that there is an induced action of all automorphisms of the pure braid groups on the homotopy groups of the two-sphere. Besides, we compute this action for a small number of strands.

Action of automorphisms of pure braid groups on homotopy groups of two-sphere

TL;DR

This work analyzes how automorphisms of the pure braid group act on the homotopy groups through the Moore complex of a Delta-group built from Brunnian braids. It proves that the subgroups and are characteristic in , inducing a well-defined action on , and provides explicit computations for small (notably is an isomorphism and has image ). The paper also gives detailed expressions for the relevant subgroups (Brun and Bd) and proves invariance under the key automorphism classes , , and . It culminates with a concrete demonstration at that the induced action is nontrivial and discusses a general conjecture that for all , along with corrections to earlier Delta-group arguments in the literature. These results connect Brunnian braids with the homotopy-theoretic structure of and illuminate how braid automorphisms modulate higher homotopy data.

Abstract

We examine the Moore complex of the Delta-group structure related to the pure braid groups and introduced by Berrick, Cohen, Wong, and Wu. We prove that the cycle and the boundary groups are invariant under all automorphisms of the pure braid groups, and thereby, we extend the results of Li and Wu on the reflection automorphism. We conclude that there is an induced action of all automorphisms of the pure braid groups on the homotopy groups of the two-sphere. Besides, we compute this action for a small number of strands.
Paper Structure (9 sections, 18 theorems, 89 equations, 6 figures)

This paper contains 9 sections, 18 theorems, 89 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Brunnian braid groups
  4. Automorphisms of pure braid groups
  5. Expressions for Lg and Lg
  6. Proof of Theorem \ref{['Theorem1']}
  7. Proof of Proposition \ref{['Proposition1']}
  8. Closing remarks
  9. Proof of Lemma \ref{['KeyIsomorphism']} and related issues

Key Result

Theorem 1.1

For any $n \ge 2$, both $\textrm{Z}_n$ and $\textrm{Bd}_n$ are characteristic subgroups of $P_n$, hence all automorphisms of $P_n$ induce automorphisms of $\pi_n(S^2)$.

Figures (6)

  • Figure 1: Pictures of geometric braids $\sigma_i$, $A_{i,j}$, and $A_{0,j}$.
  • Figure 2: Pairs of adjacent strands of $A_{0,1} A_{0,2} \ldots A_{0,n}$ bound ribbons.
  • Figure 3: Conjugation between $A_{i,j}^{-1}$ and $\chi_n(A_{i,j})$.
  • Figure 4: Alternative conjugation between $A_{i,j}^{-1}$ and $\chi_n(A_{i,j})$.
  • Figure 5: Conjugation between $A_{0,j}^{-1}$ and $\chi_n(A_{0,j})$.
  • ...and 1 more figures

Theorems & Definitions (35)

  • Theorem 1.1
  • Proposition 1.2
  • Lemma 2.1
  • Lemma 3.1
  • proof
  • Corollary 3.2
  • proof
  • Proposition 3.3
  • proof
  • Lemma 4.1
  • ...and 25 more