辫的协迫与Nielsen理论的关系

Braid theory is derived from topology and has close connection with many fields of mathematics. Nielsen fixed point theory is a classical branch in algebraic topology.In the research project,we focus our attention on the relation between the braid forcing and the Nielsen theory.By means of Nielsen theory and analyzing,we get the sufficient conditons which could distingguish the n+1-strand forcing braids of the given n-strandd braid.Thus we could estimate the number of the forcing braids.We pick up a nontrivial n-strand braid from the braid group and construct two extensions of the given braid by adding a fixed point or a periodic point of the homeomorphism.Firstly,we study the concrete forms of the two extensions.Secondly,combing with the weakly common fixed or periodic point class theory in Nielsen theory,we try to conclude a way that we could differentiate the braid forcing of the n+1-strand braid extensions.Furthermore,the number of the forcing braids becomes to the computation of the Nielsen number of the homeomorphism on the compact punched disk which has solved.Thus we get an estimation of the n+1-strand forcing btaids of the given braid.It is very important for the study of braid group theory in topology.

辫群理论起源于拓扑学，与许多数学领域都有密切联系，而Nielsen不动点理论是代数拓扑学中的经典分支.本项目中，我们主要关注于辫的协迫与Nielsen理论之间的关系，以Nielsen理论为工具，通过分析得到判别已知n-股辫的n+1-股协迫辫的充分条件，进而对协迫辫的个数给出一个估计.取定辫群中的一个非平凡n-股辫，我们利用添加此辫所对应同胚的不动点或周期点的方法，构造两类已知辫的扩展.首先，我们研究这两类辫的具体形式；其次，借助Nielsen理论中弱公共不动点类或周期点类得出如何判别n+1-股扩展辫是否为协迫辫的方法.进一步我们把协迫辫的计算转化为紧化带洞圆盘上自同胚的Nielsen数的计算问题，而关于此类同胚的 Nielsen数已经解决. 因此，便可以得到已知辫的n+1-股协迫辫个数的估计量.这对于拓扑学中辫群理论的研究有重要意义.

本项目探讨了辫群与Nielsen理论的关系，我们以Nielsen理论为工具，研究辫群中辫的协迫问题。已知一个非平凡 n-股辫，我们把辫群和已知辫所确定的同胚的映射环联系起来，通过对已知辫所确定的同胚添加不动点与周期点的方法得到两类扩展辫的具体形式，并且找出它们与映射环的基本群中元素的对应。同时利用此对应结合Nielsen理论中弱公共不动点类或周期点类的知识，得出判别n+1-股扩展辫是协迫辫的方法，进而得到n+1-股协迫辫的估计量。这样，协迫辫的个数问题也自然转化到了紧化带洞圆盘上自同胚的Nielsen数的计算问题上，为相应的研究建立了理论基础。