A theory of orbit braids

TL;DR

This paper develops a comprehensive theoretical framework for orbit braids in topological manifolds with group actions, introducing orbit braid groups, their relations, and presentations, extending classical braid theory.

Contribution

It introduces the orbit braid group $ ext{B}_n^{orb}(M,G)$, linking it to extended fundamental groups and providing presentations for specific cases, advancing the understanding of braid groups with symmetries.

Findings

Orbit braid groups contain fundamental groups of configuration spaces as subgroups.

Presented explicit generators and relations for orbit braid groups in specific cases.

Connected orbit braid groups to classical braid groups like $Br(B_n)$ and $Br(D_n)$.

Abstract

This paper upbuilds the theoretical framework of orbit braids in $M\times I$ by making use of the orbit configuration space $F_G(M,n)$, which enriches the theory of ordinary braids, where $M$ is a connected topological manifold of dimension at least 2 with an effective action of a finite group $G$ and the action of $G$ on $I$ is trivial. Main points of our work include as follows. We introduce the orbit braid group $\mathcal{B}_n^{orb}(M,G)$, and show that it is isomorphic to a group with an additional endowed operation (called the extended fundamental group of $F_G(M,n)$), formed by the homotopy classes of some paths (not necessarily closed paths) in $F_G(M,n)$, which is an essential extension for fundamental groups. The orbit braid group $\mathcal{B}_n^{orb}(M,G)$ is large enough to contain the fundamental group of $F_G(M,n)$ and other various braid groups as its subgroups. Around the central position of $\mathcal{B}_n^{orb}(M,G)$, we obtain five short exact sequences weaved in a commutative diagram. We also analyze the essential relations among various braid groups associated to those configuration spaces $F_G(M,n), F(M/G,n)$, and $F(M,n)$. We finally consider how to give the presentations of orbit braid groups in terms of orbit braids as generators. We carry out our work by choosing $M=\mathbb{C}$ with typical actions of $\mathbb{Z}_p$ and $(\mathbb{Z}_2)^2$. We obtain the presentations of the corresponding orbit braid groups, from which we see that the generalized braid group $Br(B_n)$ actually agrees with an orbit braid group and $Br(D_n)$ is a subgroup of another orbit braid group. In addition, the notion of extended fundamental groups is also defined in a general way in the category of topology and some characteristics extracted from the discussions of orbit braids are given.