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I want to know what's the fundamental group of the configuration space of closed half plane. closed half plane: $$ H_{2}\equiv\{(x_1,x_2)\in\mathbb{R}^2\mid x_2\ge0\}. $$ Then I want to consider the fundamental group of its unordered configuration space: $$ \pi_1(\text{UConf}_{n}(H_2)) $$ How can i get it?

Is this untrivial problem? Because I wanted to treat the closed half plane as the upper half plane with its boundary the real axis but found a paper Configuration Spaces of Manifolds with Boundary. Although i cannot understand it, it seems to me that this paper tells this is not a trivial problem.

Surelykidding
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  • The configurations in a manifold with boundary are homotopy equivalent to the configurations in the interior via pushing along a collar. As such, you are asking about configurations in the plane, which a lot is known about (see this question for instance). The fundamental group is given by the braid group $B_n$. – Connor Malin Aug 30 '23 at 03:27
  • @ConnorMalin Could you give me some reference about "The configurations in a manifold with boundary are homotopy equivalent to the configurations in the interior via pushing along a collar". Much thanks. I just wonder if that so why those people discussed the configuration space by dividing the problem to the closed case and with boundary case. – Surelykidding Aug 30 '23 at 03:52
  • That paper is written in order to set up theory of cobordism / handle attachments for configuration spaces which requires paying special attention to configurations with a point in the boundary. – Connor Malin Aug 30 '23 at 04:33
  • @ConnorMalin Oh! thanks. Sorry for misunderstanding the paper. Could you please give me some references so i could have a better understanding about this whole process. – Surelykidding Aug 30 '23 at 06:35
  • I would highly recommend Sinha's article on ordered configurations in the plane https://arxiv.org/abs/math/0610236 . I'm not an expert about the unordered configurations, but if you look at things related to braid groups you will find information. – Connor Malin Aug 30 '23 at 18:25

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