The Fundamental Group of $SO(n)$ Via Quotients of Braid Groups

TL;DR

This paper provides an algebraic proof that the fundamental group of the special orthogonal group SO(n) is isomorphic to Z/2Z, using quotients of braid groups and their centers.

Contribution

It introduces a novel algebraic approach to determine the fundamental group of SO(n) via quotient groups of braid groups with added relations.

Findings

The fundamental group of SO(n) is isomorphic to Z/2Z.

The fundamental group appears as the center of a finite quotient of the braid group.

The quotient group is a nontrivial central extension of the hyperoctahedral symmetry group.

Abstract

We describe an algebraic proof of the well-known topological fact that $\pi_1(SO(n)) \cong Z/2Z$. The fundamental group of $SO(n)$ appears in our approach as the center of a certain finite group defined by generators and relations. The latter is a factor group of the braid group $B_n$, obtained by imposing one additional relation and turns out to be a nontrivial central extension by $Z/2Z$ of the corresponding group of rotational symmetries of the hyperoctahedron in dimension $n$.