三维流形穿孔球面和及其在纽结补空间研究中的应用

In recent years, with the ever bettering of the research of the genus of amalgamated 3-manifolds, people pay much attention to studying the genus of the bounded surface sum of 3-manifolds. The applicant has obtained some results in estimating the genus of the bounded surface sum of 3-manifolds. In this project, the applicant takes advantage of incompressible and pairwise incompressible surfaces with meridional boundaries in knot complements to combine essential tangle decompositions of knots with the study of the punctured sphere sum of 3-manifolds and knot complements，which provides a new idea for the research on properties of knot complements. Firstly, this project discuss the classification of essential closed surfaces in the punctured sphere sum of product I-bundle of closed surfaces by using the techniques and methods of combinatorial topology of 3-manifolds, and the results can be applied to classify closed incompressible surfaces in knot complements. Secondly, we explore how to generalize the additivity of the genus of the punctured sphere sum of 3-manifolds, and estimate the genera of knot complements. Lastly, for any given knot with particular tangle decomposition, according to the topological graphs of incompressible and pairwise incompressible surfaces in knot complement, we analyze the classification of essential tori and closed incompressible, pairwise incompressible surfaces in knot complement.

近年来，随着三维流形融合积的亏格的研究越来越完善，研究三维流形带边曲面和的亏格是人们比较关注的一个问题。申请者在三维流形带边曲面和亏格的估计上取得了一些结果。在本项目中，申请者利用纽结补空间中具有子午线边界分支的、不可压缩两两不可压缩曲面，将纽结对应的本质线分解与三维流形穿孔球面和及纽结补空间的研究紧密的结合起来，这为研究纽结补空间的性质提供了一种新的思路。本项目首先利用三维流形组合拓扑方面的技巧和方法，讨论乘积流形穿孔球面和中本质闭曲面的分类问题，并将相应的结果应用到纽结补空间闭的不可压缩曲面的分类，然后探讨把三维流形穿孔球面和具有亏格可加性推广到一般情形的可能性，给出纽结补空间亏格的估计，最后利用纽结补空间中不可压缩两两不可压缩曲面对应的拓扑图，分析具有特定线分解的纽结对应补空间中本质环面、闭的不可压缩两两不可压缩曲面的分类。

三维流形中不可压缩曲面的研究一直以来都是三维流形理论研究中的重要课题和方向，确定三维流形中不可压缩曲面的存在性、性质和分类是三维流形不可压缩曲面研究领域内的三个中心问题。近年来，研究复杂三维流形沿带边曲面相粘所得流形的亏格、相粘因子流形的亏格以及相粘曲面欧拉示性数三者之间的关系也是低维拓扑的研究者们比较感兴趣的问题。本项目利用三维流形组合拓扑已有的研究方法和技巧，证明了复杂三维流形的某些带边曲面和具有亏格可加性；利用纽结的线分解理论与纽结补空间不可压缩、两两不可压曲面对应的约化拓扑图，研究了交错纽结补空间中一类本质曲面的性质。特别的，本项目证明了复杂三维流形的三穿孔球面和具有亏格可加性，将复杂三维流形平环和具有亏格可加性推广到了更加一般的情形。. 本项目得到的相关结果对于研究纽结补空间的性质和确定纽结的隧道数都具有非常积极的作用。