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

TL;DR

This paper investigates how automorphisms of pure braid groups influence the homotopy groups of the two-sphere, extending previous results and computing specific actions for small cases.

Contribution

It proves invariance of cycle and boundary groups under all automorphisms of pure braid groups and describes the induced action on homotopy groups of the two-sphere.

Findings

Cycle and boundary groups are automorphism-invariant.

Automorphisms induce actions on homotopy groups.

Explicit computations for small number of strands.

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.