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

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arxiv 1607.05876 v1 pith:H7MMKHN7 submitted 2016-07-20 math.HO math.GRmath.GT

Ina Hajdini , Orlin Stoytchev This is my paper · ORCID
classification math.HOmath.GRmath.GT
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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 $n$.

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Cited by 1 Pith paper

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  1. Topological Braiding of Bloch Eigenmodes Protected by Non-Abelian Quaternion Invariants

    quant-ph 2025-07 conditional novelty 6.0 of 10

    Bloch eigenmodes of PT-symmetric three-band topological insulators braid in sequences fixed by quaternion invariants; the effect is realized in coupled acoustic cavities with a synthetic dimension.

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