中科院数学与系统科学研究院

数学研究所

中科院华罗庚数学重点实验室

综合报告会

（Colloquium）

报告人：Prof. Song Yongjin（Inha University,Korea）

题  目：Embedding problems of Artin groups into mapping class groups

时  间：05.13(星期三), 09:30--10:30 

地  点：数学院南楼N210室

Abstract: Braid groups can be embedded in mapping class groups of surfaces in various ways, mainly because there is a braid relation between two adjacent Dehn twists. The classical Harer conjecture is about the homology triviality of the obvious embedding. In the proof of this conjecture, the categorical delooping plays a key role. Both braid groups and mapping class groups have braided monoidal category structures which gives rise to double loop space structures. The homology homomorphism induced by the Harer embedding is supposed to be trivial if it preserves Kudo-Araki-Dyer-Lashof operation. In order to show this we construct two monoidal 2-categories and functors between them which give rise to two double loop spaces and a double loop space map. This is an important example of categorical delooping technique.

There are various interesting embeddings of braid groups into mapping class groups and many of them are homologically trivial. We may extend this to the case of Artin groups.  The most interesting Artin groups are exotic type (E_6, E_7, E_8) Artin groups. Wajnryb showed that there is no geometric embedding of exotic type Artin groups into mapping class groups. We now may raise a natural question. What about the existence of nongeometric embedding? This problem is still open. It now seems that it is very hard to find an example of such an embedding. If the answer is negative, it should include an important secret in the structure of mapping class groups and 3-dimensional closed manifold.