本文選題：循環(huán)上同調(diào)　切入點：極小微分同胚　出處：《吉林大學》2016年博士論文

【摘要】：本文所關心的問題是利用光滑交叉積代數(shù)在光滑flip共軛意義下分類微分同胚.設M為一個光滑流形,α為其上極小唯一遍歷微分同胚.人們可以由此定義C*交叉積代數(shù).C*代數(shù)分類理論是當代數(shù)學的前沿理論.隨著該領域領域的發(fā)展,人們漸漸對其在微分動力系統(tǒng)分類中的應用產(chǎn)生了濃厚的興趣.然而結果并不盡如人意,究其根本,在于C*代數(shù)及其分類不變量K理論并不能反映分層結構.因此,龔貴華教授提出如下兩個問題：1是否可以利用如光滑交叉積代數(shù),反映分層結構.2是否可以利用不變量循環(huán)上同調(diào),反映1中描述的現(xiàn)象.在本文中,我們借助一些例子和具體計算說明了光滑交叉積代數(shù)就是我們需要的代數(shù),而循環(huán)上同調(diào)就是我們需要的不變量.本文主要內(nèi)容與結構安排如下：,在第一章中,我們詳細介紹了本文的研究背景.正是明顯不可能光滑flip共軛的極小唯一遍歷微分同胚誘導的C*交叉積代數(shù)C(S3)×α3Z,C(S5)×α5Z互相同構,促使我們尋求更好的代數(shù).而Elliott和龔貴華教授發(fā)現(xiàn)的例子啟發(fā)我們嘗試利用光滑交叉積代數(shù)以及循環(huán)上同調(diào)解決問題.2在第二章中,我們主要介紹一些相關概念,并利用譜序列和其他代數(shù)工具,找到了反映循環(huán)上同調(diào)分層結構的群E∞n.在第一章的最后,我們也說明了后者的計算方法.本章內(nèi)容主要基于Alain Connes和]R.Nest的工作.3第三章中,我們主要利用奇數(shù)維球面的結構,構造相關例子.首先,我們考察S2n+1上的極小唯一遍歷微分同胚αn。.很明顯,如果n≠m,αn與α。不可能光滑flip共軛.然而,我們說明了相應的C*代數(shù)在n≠m時依然有：C(S2n+1)×αnz≌C(S2m+1)×αmZ.繼而我們證明了C∞(S2l+1)αlZ的循環(huán)上同調(diào)HP1≌C (?) C擁有不同的分階結構：定理0.1這說明了如果n≠m,則相應光滑交叉積代數(shù)不同構,即G∞(S2n+1)×αnZ(?)C∞(S2m+1)×αmZ.接下來,我們構造了兩個S3×S6×S8上的極小唯一遍歷微分同胚α,β.其中α翻轉S6的定向,而β翻轉S8的定向(因此它們不可能光滑flip共軛).我們證明了相應的C*交叉積代數(shù)同構,即C(S3×S6×S8)×αZ≌C(S3×S6×S8)×βZ.然而相應的光滑交叉積代數(shù)C(S3×S6×S8)×αZ,C∞(S3×S6×S8)×βZ的HP1卻是由不同階的c直和項構成.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)均為{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)各包含一個C直和項,而其他E∞*(C∞(S3×S6×S8×αZ)為零.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)均為{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)各含一個C直和項,而其他E∞*(C∞(S3×S6×S8)×βZ)為零.這證明了光滑交叉積代數(shù)C∞(S3×S6×S8)×αZ(?)C∞(S3×S6×S8)×αZ.4在第四章中,我們首先利用循環(huán)上同調(diào)證明了T2上的兩個光滑自同胚互相不光滑flip共軛.然后,我們我們計算了Furstenberg變換誘導的光滑交叉積代數(shù)的循環(huán)上同調(diào)：定理0.4因此,這樣我們就計算出Furstenber變換誘導光滑交叉積代數(shù)的循環(huán)上同調(diào)理論.
[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，

本文編號：1686903