拓扑量子场论在3维流形的映射度问题中的应用

The problem of degrees of mappings between manifolds, consisting of the existence of non-zero degree maps and description of sets of mapping degrees, is a classical topic in algebraic topology and geometric topology. It has been widely studied during the recent years and many results have been obtained, but there still remains many interesting problems. This program plans to apply a young theory-topological quantum field theory (TQFT), to the problem of mapping degrees. Given a hyperbolic 3-manifold N, one can construct a TQFT to study the set D(M,N) of degrees of mappings from M to N. We shall detailedly investigate the structure of this TQFT, and get a general method of computing D(M,N). When M is a Seifert 3-manifold or some other simple manifold, we will concretely describe D(M,N). Early work indicates that this method is successful when applied to degrees of mappings to topological spherical space forms, thus it can be expected to lead to several good results. Furthermore, in running this program, it will promote us to deepen the understanding to a class of topological quantum field theories.

流形之间的映射度问题, 包括非零度映射的存在性和映射度集的描述等方面, 是代数拓扑和几何拓扑领域的经典问题. 近年来人们对它广为研究, 并取得了丰硕的成果, 但仍有很多有意义的问题未获解答..本项目把拓扑量子场论(TQFT)这一崭新的理论应用于映射度问题. 给定3维双曲流形N, 为确定一般3维流形M到N的映射的度的集合D(M,N), 可建立一个TQFT. 我们将详尽地探讨该TQFT的结构, 得出计算D(M,N)的一般方法. 对于Seifert流形及其他一些简单的3维流形M, 将具体地描述D(M,N). 前期工作表明, 这个方法应用在流形到拓扑球空间形式的映射度上很成功, 因此可以期待取得若干好结果. 另外, 本项目的研究过程中也将促进对一类拓扑量子场论的深入了解.

本项目旨在用无限离散群建立同伦量子场论, 以应用于解决任意3维流形到双曲流形的映射度问题. 我们成功地把有限群的Dijkgraaf-Witten理论中的部分结构推广到任意群, 初步建立了cycle-cocycle运算, 为一般的Chern-Simons不变量的研究建立了基础. 为构造到双曲流形的连续映射, 我们尝试寻找纽结/链环的基本群到线性群的同态, 已经得到较多具体结果, 并为后续研究积累了丰富的经验.