三维流形融合与链环群单纯结构的若干问题研究

The three-manifold theory is a significant research area of low-dimensional topology. With the introduction of Heegaard distance, the research on amalgamations of 3-manifolds has achieved fruitful results in the past few decades, but there are still many basic unsolved questions. The main object in this project is to study the properties of essential bounded surfaces in 3-manifolds, properties of amalgamations of 3-manifolds along bounded surfaces, and the structure of link group simplicial model. The research includes: studying the topological information reflected by the essential bounded surfaces in 3-manifolds; finding the sufficient conditions for the Heegaard splittings of amalgamations of 3-manifolds along bounded surfaces to be unstabilized, critical and the variations of Heegaard genera; studying the structure of the simplicial group model obtained from framed link groups, building the algorithm to detect the unknot. The project makes use of the subsurface projection to study the intersections of essential bounded surfaces and Heegaard surfaces in 3-manifolds, and discusses the lower central series of the simplicial groups obtained from framed link groups. The results of this project will conduce to the study of essential bounded surfaces in 3-manifolds and the Heegaard splittings of amalgamations of 3-manifolds along bounded surfaces, and will also have important theoretical significance for the interdisciplinary research of homotopy theory and knot theory.

三维流形理论一直是低维拓扑学的重要研究领域。近年来，随着Heegaard距离等概念的引入，三维流形融合的相关问题取得了丰硕的研究成果，但是该领域仍有大量问题亟待解决。本项目主要研究带边三维流形中本质带边曲面的性质，三维流形沿带边子曲面融合问题，以及链环群单纯结构。研究内容包括：发掘三维流形中本质带边曲面所蕴含的拓扑信息；给出三维流形沿带边子曲面融合后Heegaard分解稳定化、critical的充分条件以及确定亏格的变化规律；研究由标架链环群所得单纯群模型的结构，建立平凡结探测算法。主要研究方法是利用子曲面投影技术对三维流形中带边本质曲面与Heegaard曲面间的相交形式进行探讨，利用降中心序列推进标架链环群单纯模型的深入研究。研究结果对于深入了解三维流形中本质带边曲面的性质，三维流形沿带边子曲面融合后Heegaard分解的性质以及同伦论和纽结论的深入交叉有重要的理论意义。

三维流形理论一直是低维拓扑学的重要研究领域。近年来，三维流形融合的相关问题取得了丰硕的研究成果，但是该领域仍有大量问题亟待解决。本项目从多角度探索三维流形的拓扑结构，在柄体曲面和、三维流形自融合的Heegaard分解结构以及三维流形把柄添加的性质方面开展了系统而深入的研究。方法上，本项目将组合拓扑和代数拓扑的方法相结合，探索三维流形的相关拓扑结构。主要研究结果包括：从代数拓扑角度给出了柄体曲面和仍为柄体的充要条件；给出了三维流形沿本质子曲面自融合之后的分解是非稳定化和uncritical的充分条件；研究了三维流形把柄添加的性质。本项目的获得的主要结果对于更多的了解低维流形的拓扑结构有重要的理论意义。