【作者】 马继明；

【导师】 廖公夫； 邱瑞锋；

【作者基本信息】 吉林大学 ， 基础数学， 2005， 博士

【摘要】 本文主要通过组合方法来探讨三维流形中的一些问题,既解结数,不可压缩曲面和Heegaard分解。主要结果如下: 1.构作了三维球面中纽结基本群的广义Wirtinger表示,利用其给出纽结的解结数这一纽结几何不变量的一个代数上的下界。 2.在一个亏格为2的可定向闭曲面的平凡Ⅰ-bundle里构作一个分支数为3的链环,并证明其补空间中含有任意正整数亏格的分离可定向不可压缩闭曲面,从而得到定理:在任意一个可定向三维流形中、存在一个分支数至多为4的链环,使得其补空间中含有任意正整数亏格的分离可定向不可压缩闭曲面。并分析了上述构造,给出了沿着亏格为2n的分离不可压缩闭曲面作非稳定化的Heegaard分解的融合积可以稳定化2n-4次的例子。 3.研究了边界可约流形的Heegaard分解,定义了“边界连接和”与“自边界连接和”的概念,证明了Heegaard分解在“边界连接和”与“自边界连接和”这两种运算下为稳定化的当且仅当在这两种运算之前的Heegaard分解是稳定化的,并深入研究了边界可约流形的Heegaard分解,证明了某种意义下的唯一性。更多还原

【Abstract】 Since the definition of the term "manifold", people have done many work on it, and founded some new branches of mathematics such as algebraic topology, differential topology and differential geometry. But in dimension three, Whitney trick and character class are almost useless, so some standard methods in high dimensional manifolds is not in effect. People founded some new technique to study 3-manifolds, such as the combinatorial methods in 3-manifolds, differential geometry used in Geometrization Conjucture. Heegaard-Floer homology, and so on. Group theory, incompressible surfaces and Hee-gaard splittings are important combintorial methods for the study of 3-manifolds. From the work of Hempel, Birman, et al, we know that the Heegaard splitting of 3-manifolds are tying tightly to the mapping class group of surfaces and the geometric strcture of 3-manifolds.Lickorish had shown that for any orientable closed 3-manifold M, there is a link L in S~3. an surgery on L, such that the resulting manifold is M. H. Hilden and J. Montesinos used different methods shown that for any orientable closed 3-manifold M, there is a knot K in S~3, such that M is the branched covering of S~3 with the branched set K. C. A. Gordon and J. Luecke in 1989 proved that the knots K in S~3 is determined by there complements. As all above, the study of 3-manifolds and knot are tied tightly.Since the high dimensional homotopy groups and homology groups of thecomplements of knots in S3 are trivial, so the 1-dimensional homotopy groups, that is, the fundamental groups of the complements of knots in S3, are so important. This group is finitely presentation group, and one can give a finitely presentation of it from the braid presention of the knot, and other methods to obtain the finitely presentation of it, different finitely presentations have different effect on the study. In fact, Wirtinger gived a finitely presentation from a projection diagram of a knot, and in this thesis, we shall give a modified Wirtinger presentation, and the application to the unknotting number is touched. The unknotting number of a knot, is a geometric invariant, the numeration of it is so difficult, so people gave many estimations of it. The lower bound we gived is from algebraic viewpoint, al thro ugh it is not accurate, but we wish it can help the study of the unknotting number, such as about composite knots.For the study of 3-manifolds, we can first study the 2-manifolds embedded in the 3-manifolds, that is, surfaces. We must study the embedded 2-manifolds which satisfy some conditions, such as, incompressible surfaces, Heegaard surfaces, etc.Incompressible surface is just the properly embedded surface that the induced map on the fundamental groups is injective, it has many applications, such as Thurston’work on the geometric structure of Haken manifolds. It is in-terseting and useful to construct incompressible surfaces in 3-manifolds. There are some works, for example, W. Jacobs work on nonseparating incompressible surfaces with boundary in handlebodies, and R. F. Qiu’s work on separating incompressible surfaces with boundary in handlebodies. This thesis generalizes the result of T. Kobayashi, R. F. Qiu, Y. Rieck and S. C. Wang, so we can construct arbitrary positive genus closed incompressible surfaces in some3-manifolds.Heegaard surface, roughly speaking, is the surface that can "compress completely" , and it is corresponding to the Heegaard splitting, the important combinatorial structure, of the 3-manifold. Thanks to Heegaard, Moise, etc, every 3-manifold has Heegaard splittings, so Heegaard splitting is a unitive method to construct 3-manifolds. But every 3-manifold has Heegaard splittings of arbitrary high genus, so we must study the relations between different Heegaard splittings, and have the term "reducible", "boundary-reducible", "stabilization", etc.In 2004, R. F. Qiu gave a proof of the famous Gordon Conjecture, that say, a reducible Heegaard splitting is stabilized iff one of the factors is stabilized. And then we have the generalized Gordon Conjecture, that is, the amagalmation of two unstabilized Heegaard splitting along a genus g closed incompressible surface can stabilize at most 2g times. On the considering of the generalized Gordon Conjecture, we prove the disk version’s Gondon Conjecture, have some interesting and important results, and so we can study the Heegaard splitting of boundary-reducible 3-manifolds in-depth.The unknotting number of a knot(just link with only one component) is an interesting and important invariant, in this thesis, we use group theory and give a lower bound of it:1. We give a modified Wirtinger presentation of the fundamental group 7a (S3 - K) of knot complement K in S3.2. we define an knot invariant a(K), and show that it is a lower bound of the unknotting number u(K).Generalizing the work of T. Kobayashi, R. F. Qiu, Y. Rieck and S. C. Wang, we construct a set of link complements M, and show that M containsall positive genus closed separating incompressible surface, then we study the amalgamation of Heegaard splittings along the incompressible surfaces:3. If F is a genus 2 closed orientable surface, then there is a link L = Ki U K2 U K3 in F x [0,1], such that (F x [0,1])L contains all positive genus closed incompressible surfaces.4. For any orientable 3-manifold M, there is a link Lin M, \ L |< 4, such that ML contains all positive genus closed separating incompressible surfaces.5. there is a manifold M, which contains genus 2n separating closed incompressible surfaces F, F separates M into M\ and M2} Mi has unstabilized Heegaard splitting H\ U H%2, such that when amalgamating H[ U H\ along F, the resulting Heegaard splitting can stabilize 2n-4 times.In fact, the result of T. Kobayashi, R. F. Qiu, S. C. Wang and Y. Rieck gave the example that the amagalmation of two unstabilized Heegaard splitting along a genus 2g + 1 closed incompressible surface can stabilize 2# — 2 times. Our example and T. Kobayashi, R. F. Qiu, S. C. Wang, Y. Rieck’s example although are not the best examples on the superior bound of the generalized Gordon Conjecture, but are the only ones on this problem.A fundamental problem on Heegaard splittings is stabilization. In this thesis, we define "boundary-connected sum" and "self-boundary-connected sum" of Heegaard splittings, and prove the followings:6. ’boundary-connected sum’" of Heegaard splittings is stabilized iff one of the factor is stabilized.7. Doing a "self-boundary-connected sum" along M’ = W’ U V’, the resulting Heegaard splitting is stabilized iff M’ = W’ U V’ is stabilized.8. 1) Any Heegaard splitting of a 9-reducible manifold M, say M = H^U V, can be obtained by doing "connected-sums", "boundary-connected-sums"更多还原