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Segal's electrostatic map given in "Configuration Spaces and Iterated Loop...
In Segal's "Configuration Spaces and Iterated Loop Spaces", I'm not understanding the map $C_n(X) \rightarrow \Omega^n S^n X$ given in the first page, Section 1 (the picturesque electrostatic map) as...
View ArticleExtension between the abelianization of the pure braid group and the...
The braid group $B_n$ and the pure braid group $P_n$ sits in a short exact sequence$$1\to P_n\to B_n\to S_n\to 1.$$The pure braid group $P_n$ has abelianization $\mathbb Z^{n\choose 2}$, with...
View ArticleExample from Segal's "Configuration Spaces and Iterated Loop Spaces
After Theorem 3 in Segal's Configuration Spaces and Iterated Loop Spaces, he gives some special cases.I do not understand $n = 2$ case, i.e. how is $B(\coprod_{k\geq 0} B(Br_k)) \simeq \Omega...
View ArticleMarkov theorem for braid partial closures
The classical Markov theorem tells us that the closures of two braids are isotopic links if and only if the braids are related by a sequence of Markov moves(MI) $b \sim aba^{-1} $(MII) $b \sim b...
View ArticleWhat do tangles teach us about braids?
A braid is a smooth level-preserving embedding $f\colon\, \{1,2,\dotsc,n\}\times[0,1]\hookrightarrow \mathbb{R}^2 \times [0,1]$ such that $f(k,0)=(k,0)$ and $f(k,1) \in \{1,2,\dotsc,n\} \times \{1\}$....
View ArticleMapping class group of $n$-punctured annulus
I am looking for an explicit presentation of the mapping class group of the annulus $\mathbb{A}^2$, after equipping it with $n$ interior punctures/marked points $\{x_1, \cdots, x_n\} \hookrightarrow...
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