Vector Braids

TL;DR

This paper introduces a new family of groups called vector braids, generalizing classical braid groups to points in complex spaces and projective spaces, and explores their algebraic and homological properties.

Contribution

It defines the vector braid groups, provides presentations, and investigates their homology, extending classical braid theory to higher-dimensional and projective settings.

Findings

Presented a group $PL_n$ surjecting onto $P_n^2$

Showed $PL_n$ induces isomorphism on first and second homology with $P_n^2$

Derived an infinitesimal presentation of $P_n^2$

Abstract

In this paper we define a new family of groups which generalize the {\it classical braid groups on} $\C $. We denote this family by $\{B_n^m\}_{n \ge m+1}$ where $n,m \in \N$. The family $\{ B_n^1 \}_{n \in \N}$ is the set of classical braid groups on $n$ strings. The group $B_n^m$ is the set of motions of $n$ unordered points in $\C^m$, so that at any time during the motion, each $m+1$ of the points span the whole of $\C^m$ as an affine space. There is a map from $B_n^m$ to the symmetric group on $n$ letters. We let $P_n^m$ denote the kernel of this map. In this paper we are mainly interested in understanding $P_n^2$. We give a presentation of a group $PL_n$ which maps surjectively onto $P_n^2$. We also show the surjection $PL_n \to P_n^2$ induces an isomorphism on first and second integral homology and conjecture that it is an isomorphism. We then find an infinitesimal presentation of the group $P_n^2$. Finally, we also consider the analagous groups where points lie in $\P^m$ instead of $\C^m$. These groups generalize of the classical braid groups on the sphere.