分野別セミナー

アブストラクト：
Abstract: In 1891, Hurwitz introduced the (pure) braid group as the fundamental group of the configuration space of points in the complex plane. Later, in 1925, Artin studied its algebraic and geometric perspectives. In particular, he gave a presentation of the braid groups, which is now recognized as a standard one. He also studied the pure braid group and gave its presentation. However, it is much more complicated than the one of the braid group. Thus, there is nothing better than to have a simple to look and intuitive presentation of the pure braid group. In my talk, even though there's no new presentation, we connect two different approaches to the pure group: from group theory and from hyperplane arrangement. Firstly, we introduce a "lexicographic section'' of the braid arrangement, and give a presentation of the fundamental group of its complement using the braid monodromy technique. Then, show that the resulting presentation coincides with the modified Artin presentation given by Margalit--McCammond. If I have time, I also generalize the construction to the Manin--Schechtman arrangement MS(n,k), a generalization of the braid arrangement. In particular, we give a presentation of π_1(C^5 \ MS(5,2)), which is the simplest and nontrivial example of the Manin--Schechtman arrangements.

URL: https://sites.google.com/view/mathpolynomial