O grupo das tranças no disco unitário como aplicação de grupos quocientes

Abstract

The Braid Group Over the Unit Disk (Braid Groups) has 3 different equivalent definitions thanks to Mathematical different structures: as quotient groups ([L]) over Abstract Algebra ([H, M]), as fundamental group of configuration spaces and also, as mapping class groups; this last two structures are from heavier tools over Algebraic Topology ([B]). Here we present The Braid Group over the Unit Disk, or simply, Artin Braid Groups as is also known, as a application of a special construction under quotient groups: we consider a set (of braids on n-strands), quotient up to a equivalence relation of elementary movements (formally called ambient isotopy) and finally, equip this quotient set with a product operation (concatenation), giving rise to the mentioned group above. Furthermore, we discuss over the finitely presentation for The Braid Group Over the Unit Disk given by Artin in 1925. This presentation became known worldwide only in 1946, when the paper was translated to English in that time ([A1, A2]). Finally, we talk about a famous problem on conjugacy theory, called Word Problem, which consists in providing an algorithm which decides when a given braid is equivalent to the trivial braid or not. The approach we suggest here is given in a geometrical way as is done in [M], since is a manner to undergraduate students are able to understand, as much difficult the theory is. This work was a result from years of scientific cooperation between the student and her advisor.