Braid group

Let $𝑋$ be a topological space, and $(𝑋 𝑛)$ denote the space of subsets of $𝑋$ of cardinality $𝑛$ .¹ The braid group $𝔅 𝑛 (𝑋)$ on $𝑛$ strands in $𝑋$ is the fundamental group

𝔅 𝑛 (𝑋) = 𝜋 1 (𝑋 𝑛) .

In the special case $𝑋 = ℝ 2$ is the Euclidean plane, we have the Artin braid group.

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The natural topology on $(𝑋 𝑛)$ is that of the orbit space (quotient topology) of [[Symmetric group| $S 𝑛$ ]] acting on the subspace topology of the product space $𝑋 𝑛 ∖ Δ$ where $Δ$ consists of points with at least two components the same. ↩

Braid group

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Braid group

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