On the Contribution of Wu Wen-Tsün to Algebraic Topology

论文摘要

The aim of this article is to present the contribution of Wu Wen-Tsün to Algebraic Topology and more precisely to the theory of characteristic classes. Several papers provide complete and welldocumented biography and academic career of Wu Wen-Tsün, in particular, Hudecek, 2014; O’Connor and Robertson, 2006; Wen-Tsün Wu’s Academic Career, 2006; Selected works of Wen-Tsun Wu, 2008.The author does not repeat the details provided in these papers concerning the Wu Wen-Tsün’s bibliography, we will just mention people involved in the Wu Wen-Tsün’s period in France.In addition to Wu Wen-Tsün’s papers, the Dieudonné’s book（Dieudonné, 1960） provides an excellent presentation of main results of Wu Wen-Tsün in Algebraic and Differential Topology. The author will use and abuse of this book（and refer to） when suitable.In the introduction, the author recalls mainly historical facts concerning the contribution of Wu Wen-Tsün to Algebraic Topology. The second section shows speci?cally the contribution of Wu WenTsün to the Stiefel-Whitney classes and introduces the third section, dealing with the（real） Wu classes.The author provides de?nition, properties as well as further developments and generalizations of the Wu classes. The fourth and ?fth sections are devoted to recent applications: In Cobordism theory and in Mathematical Physics. The author notices that Wu classes have been used as well in other domains,in particular surgery theory（Madsen and Milgram, 1979）. The last section concerns the complex Wu classes and shows that the more recent Mather classes coincide with the previously de?ned complex Wu classes, that is a result from Zhou（1994）（see also Liu, 1996）.This article is devoted to the contribution of Wu Wen-Tsün to the theory of Characteristic Classes,which coincides with his "French period"（1947–1951）. However, speaking of Algebraic Topology, it is worthwhile to mention the important contribution of Wu Wen-Tsün to the Theory of realization of complexes or manifolds in Euclidean spaces and of embedding classes. That coincides with his return to China（1956–1965）.

论文目录

• 1 Introduction—A Few of History
• 2 Stiefel-Whitney Classes
•   2.1 Obstruction Theory
•   2.2 Grassmannian
•   2.3 Axiomatic Definition of the Stiefel-Whitney Classes
• 3 Wu Classes
•   3.1 Steenrod Squares
•     3.1.1 i-Products
•     3.1.2 Product of Steenrod Squares
•     3.1.3 Axiomatic Definition of the Steenrod Squares
•   3.2 Stiefel-Whitney Class and Thom Class
•   3.3 Wu Classes
•   3.4 Siefel-Whitney Classes and Wu Classes
•   3.5 Further Properties and Generalizations
•     3.5.1 Additivity
•     3.5.2 Odd Dimensional Wu Classes
•     3.5.3 A Wu Formula in Zp
•     3.5.4 Atiyah-Hirzebruch generalization
•     3.5.5 Yoshida-Stong Formulae
•     3.5.6 Index of 4n Dimensional Manifolds
•     3.5.7 Relation to Pontrjagin Classes
• 4 Applications of Wu Classes-1.Cobordism Theory
•   4.1 The Thom Theorem
•   4.2 The Singular Case-Intersection Homology
•     4.2.1 Pseudomanifolds
•     4.2.2 Intersection Homology and Cohomology
•   4.3 The Singular Case-Cobordism of Spaces
•   4.4 The Singular Case-Cobordism of Maps
• 5 Applications of Wu Classes-2.Mathematical Physics
• 6 The Complex Framework:Chern Classes and Wu Complex Classes
•   6.1 Chern Classes of Vector Bundles
•   6.2 Chern Classes of Singular Varieties
•   6.3 Wu Transformation
•   6.4 Wu Classes （Complex Case）
•   6.5 Mather Classes
•   6.6 Wu Classes and Mather Classes

文章来源

类型: 期刊论文

作者: BRASSELET Jean-Paul

来源: Journal of Systems Science & Complexity 2019年01期

年度: 2019

分类: 基础科学

专业: 数学

单位: I(CNRS)-Aix-Marseille University

分类号: O189.2

页码: 3-36

总页数: 34

文件大小: 508K

下载量: 6