The Fundamental Group of $SO(n) Via Quotients of Braid Groups
We describe an algebraic proof of the well known topological fact that \pi_1(SO(n) \cong Z/2Z$ 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 $n$