基于切片2-复形的Andrews-Curtis不变量的研究
Considerations about Andrews-Curtis invariants based on sliced 2-complexes
摘要：
我们考虑一个特殊形式的2-复形，称为2-复形的奎因模型。它可以被划分成图，从一个图到另一个图的变化可以通过一系列局部转换来组织，这些转换在F的列表中描述。Quinn [Q1].将2-复形分解为图必须转换为代数背景（例如拓扑量子场论（TQFT）），以构造3-变形下合适的势不变量。这些不变量可以通过使用超级计算机进行计算，其结果可能会产生Andrews-Curtis猜想的反例。为了实现3-变形下的不变性，局部变换之间存在明显的拓扑关系，例如将气泡变形出矩形。在本文中，我们的主要结果是，我们提供了一个完整的列表，这样的拓扑关系在一个完全几何的方式。我们考虑的一个结果是，F. Quinn [Q1]通过一个额外的关系进行扩展，该关系负责局部改变切片。到目前为止，我们还不知道这种关系是否是其余关系的结果。但它可能是至关重要的进一步的工作集中在这样的微妙之处，作为代数“简化”，在这里这个问题是绕过，到目前为止一直无法区分简单的同伦和3-变形。在我们的引言中，我们总结了一些已知的结果的情况下，当传递到代数;并在§ 8中，我们计算一个例子的代数TQFT，以证明我们的额外关系成立。所有的考虑都是针对具有两个生成元和两个定义关系元的2-复形进行的。但结果在一般情况下也成立。
英文摘要：
We consider a 2-complex in a particular form, called the Quinn model of a 2-complex. It can be sliced in graphs, where a change from one graph to another can be organized by a sequence of local transitions, which are described in a list of F. Quinn [Q1]. The decomposition of that 2-complex into graphs has to be translated into an algebraic context (for example Topological Quantum field theory (TQFT)) to construct suitable potential invariants under 3-deformations. These invariants are accessible for computation by using a supercomputer and the results may yield a counterexample to the Andrews-Curtis conjecture. To achieve invariance under 3-deformations, there are obvious topological relations among the local transitions, for example to deform a bubble out of a rectangle. In this paper our main result is that we contribute a complete list of such topological relations in a totally geometric fashion. One outcome of our considerations is that the corresponding list of F. Quinn [Q1] is extended by an additional relation which takes care of locally changing a slicing. We do not know so far whether this relation is a consequence of the remaining ones. But it may be crucial for further work to focus on such subtleties, as algebraic "simplifications", where this question is bypassed, so far have been unable to distinguish between simple homotopy and 3-deformations at all. In our introduction we summarize some known results on the situation when passing to Algebra; and in § 8 we calculate an example of an algebraic TQFT in order to demonstrate that our additional relation holds. All considerations are carried out for 2-complexes with two generators and two defining relators. But the results also hold in the general case.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
