Towards an Algebraic Characterization of 3-dimensional Cobordisms
Towards an Algebraic Characterization of 3-dimensional Cobordisms
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3 维配边的代数表征
DOI：
10.1090/conm/318/05549
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发表时间：
2001
期刊：
影响因子：
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通讯作者：
T. Kerler
中科院分区：
文献类型：
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作者：
T. Kerler
本文的目的是寻找3-流形在Hopf代数表达式上的接近同构表示。为此，我们定义并比较了Hopf代数和三维拓扑研究中自然出现的三种不同的编织张量类别。第一种是具有一个边界分量和三维相对协数的连通曲面的范畴Cob，第二种是具有关系的缠结的范畴Tgl，第三种是由Hopf代数对象自由生成的自然代数范畴Alg。从以前的工作中我们知道Tgl和Cob是等效的。我们利用这个事实和heegard分裂的思想构造了一个从Alg到Cob的满活动函子。我们还找到了一个映射，该映射与Alg中的Cob preimages中的映射类组的生成器相关联。为这些表达式验证映射类组中的单个块关系。我们建议找到一个可能具有附加关系的Alg版本，以获得映射类群的同构代数表示，并最终获得Cob的同构代数表示。1
The goal of this paper is to find a close to isomorphic presenta tion of 3-manifolds in terms of Hopf algebraic expressions. To this end we define and compare three different braided tensor categories that arise naturally in the study of Hopf algebras and 3-dimensional topology. The first is the category Cob of connected surfaces with one boundary component and 3-dimensional relative cobordisms, the second is a category Tgl of tangles with relations, and the third is a natural algebra ic category Alg freely generated by a Hopf algebra object. From previous work we know that Tgl and Cob are equivalent. We use this fact and the idea of Heegaard splittings to construct a surje ctive functor from Alg onto Cob . We also find a map that associates to the generators of the mapping class group in Cob preimages in Alg . The single block relations in the mapping class group are verified for th ese expressions. We propose to find a version of Alg with possibly additional relations to obatin isomorphic al gebraic presentations of the mapping class groups and eventually of Cob . 1