关于三维流形Heegaard分解的球面复形及其他复形的研究

Heegaard分解理论是三维流形拓扑的重要组成部分，近年来发展起来的曲线复形理论也有着广泛的应用。在曲线复形之后，更多的关于曲面的单纯复形和关于柄体的单纯复形出现了，这些复形对于研究曲面的性质和柄体的性质都起到了重要作用。本项目以三维流形的Heegaard分解理论为基础，借助组合手段，进一步研究复形理论在三维流形Heegaard分解中的应用。主要内容有：借助特殊曲面构造三维流形Heegaard分解的更高维距离并研究其性质，研究与柄体相关的特殊复形的自同构群与柄体的映射类群及柄体边界曲面的映射类群之间的关系，用组合方法证明三维球面的高亏格Heegaard分解中可约2-球面复形的连通性等性质，并构造其他Heegaard分解中可约2-球面复形具有连通性的三维流形。项目研究结果对于刻画三维流形Heegaard分解的性质有重要的意义。

The theory of Heegaard splitting is an important part of 3-manifold theory, and the curve complex theory developed recently has been widely used. After the definition of curve complex, more simplicial complexes of surface and handlebody appear, which play an important role in studying the properties of surface and handlebody. Based on the Heegaard splitting of 3-manifold, this project further studies the application of complex theory in Heegaard splitting of 3-manifold by means of combination method. The main contents include constructing the high-dimensional distances of Heegaard splitting with special surfaces and studying their properties, studying the relationship between the automorphism group of the complexes of handlebody and the mapping class group of handlebody and the mapping class group of the boundary surface of handlebody, proving the connectedness of reducible 2-sphere complex in high genus Heegaard splitting of 3-sphere combinatorially, and constructing the 3-manifolds whose Heegaard splittings having connected reducible 2-sphere complex. The results of this research are significant to describe properties of Heegaard splittings of 3-manifolds.