On homotopy aspects and cablings of virtual pure braid groups

TL;DR

This paper investigates the homotopy properties and cabling structures of virtual pure braid groups, revealing new connections to the homotopy groups of the 3-sphere and the theory of Brunnian virtual braids.

Contribution

It introduces a novel analysis of the simplicial structure of virtual pure braid groups and relates their structure to homotopy groups and cabling operations.

Findings

VP_n for n > 4 is determined by VP_3, VP_4, and virtual cablings.

New connections between virtual braid groups and homotopy groups of the 3-sphere.

Framework for understanding virtual pure braid groups via simplicial and cabling structures.

Abstract

By exploring simplicial structure of pure virtual braid groups, we give new connections between the homotopy groups of the 3-sphere and the virtual braid groups that are related to the theory of Brunnian virtual braids. The group structure of VP_n with n > 4 is determined by VP_3, VP_4 and virtual cablings given by iterated degeneracy operations on the generators and defining relations. The complete proofs will be published in the two forthcoming papers.