循环上同调与极小唯一遍历动力系统

发布时间：2018-03-30 17:31

  本文选题：循环上同调　切入点：极小微分同胚　出处：《吉林大学》2016年博士论文

【摘要】：本文所关心的问题是利用光滑交叉积代数在光滑flip共轭意义下分类微分同胚.设M为一个光滑流形,α为其上极小唯一遍历微分同胚.人们可以由此定义C*交叉积代数.C*代数分类理论是当代数学的前沿理论.随着该领域领域的发展,人们渐渐对其在微分动力系统分类中的应用产生了浓厚的兴趣.然而结果并不尽如人意,究其根本,在于C*代数及其分类不变量K理论并不能反映分层结构.因此,龚贵华教授提出如下两个问题：1是否可以利用如光滑交叉积代数,反映分层结构.2是否可以利用不变量循环上同调,反映1中描述的现象.在本文中,我们借助一些例子和具体计算说明了光滑交叉积代数就是我们需要的代数,而循环上同调就是我们需要的不变量.本文主要内容与结构安排如下：,在第一章中,我们详细介绍了本文的研究背景.正是明显不可能光滑flip共轭的极小唯一遍历微分同胚诱导的C*交叉积代数C(S3)×α3Z,C(S5)×α5Z互相同构,促使我们寻求更好的代数.而Elliott和龚贵华教授发现的例子启发我们尝试利用光滑交叉积代数以及循环上同调解决问题.2在第二章中,我们主要介绍一些相关概念,并利用谱序列和其他代数工具,找到了反映循环上同调分层结构的群E∞n.在第一章的最后,我们也说明了后者的计算方法.本章内容主要基于Alain Connes和]R.Nest的工作.3第三章中,我们主要利用奇数维球面的结构,构造相关例子.首先,我们考察S2n+1上的极小唯一遍历微分同胚αn。.很明显,如果n≠m,αn与α。不可能光滑flip共轭.然而,我们说明了相应的C*代数在n≠m时依然有：C(S2n+1)×αnz≌C(S2m+1)×αmZ.继而我们证明了C∞(S2l+1)αlZ的循环上同调HP1≌C (?) C拥有不同的分阶结构：定理0.1这说明了如果n≠m,则相应光滑交叉积代数不同构,即G∞(S2n+1)×αnZ(?)C∞(S2m+1)×αmZ.接下来,我们构造了两个S3×S6×S8上的极小唯一遍历微分同胚α,β.其中α翻转S6的定向,而β翻转S8的定向(因此它们不可能光滑flip共轭).我们证明了相应的C*交叉积代数同构,即C(S3×S6×S8)×αZ≌C(S3×S6×S8)×βZ.然而相应的光滑交叉积代数C(S3×S6×S8)×αZ,C∞(S3×S6×S8)×βZ的HP1却是由不同阶的c直和项构成.定理0.2Hcoeq0eq(S3×S6×S8,α)=CHcoeq(S3×S6×S8,α)=C, Heq3(S3×S6×S8,α)=CHeq11(S3×S6×S8,α)=C,其余所有Heq*(S3×S6×S8,α)和Hcoeq*(S3×S6×S8,α)均为{0}.即E∞1(C∞(S3×S6×S8×αZ) E∞3(C∞(S3×S6×S8×αZ) E∞9(C∞(S3×S6×S8×αZ) E∞11(C∞(S3×S6×S8×αZ)各包含一个C直和项,而其他E∞*(C∞(S3×S6×S8×αZ)为零.定理0.3Hcoeq0(S3×S6×S8,β)=C,coeq6(S3×S6×S8,β)=C, Heq3(S3×S6×S8,β)=C,Heq9(S3×S6×S8,β)=C,所有其他Heq*(S3×S6×S8,β)和Hcoeq*(S3×S6×S8,β)均为{0}.也即E∞1(C∞(S3×S6×S8)×βZ) E∞3(C∞(S3×S6×S8)×βZ) E∞7(C∞(S3×S6×S8)×βZ) E∞9(C∞(S3×S6×S8)×βZ)各含一个C直和项,而其他E∞*(C∞(S3×S6×S8)×βZ)为零.这证明了光滑交叉积代数C∞(S3×S6×S8)×αZ(?)C∞(S3×S6×S8)×αZ.4在第四章中,我们首先利用循环上同调证明了T2上的两个光滑自同胚互相不光滑flip共轭.然后,我们我们计算了Furstenberg变换诱导的光滑交叉积代数的循环上同调：定理0.4因此,这样我们就计算出Furstenber变换诱导光滑交叉积代数的循环上同调理论.
[Abstract]:This concern is the use of smooth crossed product algebra in the sense of flip conjugate smooth diffeomorphism classification. Let M be a smooth manifold, the minimal alpha is uniquely ergodic homeomorphism. One can define C* bicrossproduct.C* algebra is the classification theory of contemporary mathematics theory. Along with the development of the field the people began to have a strong interest in the application of differential dynamic system classification. However, the results are not satisfactory. The reason is that C* algebra and its classification invariant K theory does not reflect the hierarchical structure. Therefore, Professor Gong Guihua proposed the following two questions: 1 whether it can be used as smooth crossed product algebras. Reflect the hierarchical structure of.2 can use the invariants of cyclic cohomology, reflect the phenomenon described in 1. In this paper, we use some specific examples and shows smooth cross product We need algebra is algebraic, and cyclic cohomology we need is invariant. The main contents of this paper, and the structure is as follows: in the first chapter, we introduce the research background of this paper. It is obviously impossible to cross C* product algebras C smooth flip conjugate minimal ergodic diffeomorphism induced (S3) * alpha 3Z, C (S5) * alpha 5Z each isomorphism, impels us to seek a better algebra. And Professor Elliott and Gong Guihua found that the example inspired us to try to use the smooth cross product algebra and cyclic cohomology to solve the problem of.2 in the second chapter, we mainly introduce some related concepts, and the use of spectral sequences and other algebraic tools find, reflect the hierarchical structure of cyclic cohomology group E ~ n. at the end of the first chapter, we also illustrate the calculation method of the latter. This chapter is mainly based on Alain Connes and]R.Nest.3 third chapter 涓，

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