Differential Algebraic Topology: From Stratifolds to Exotic Spheres
Differential Algebraic Topology: From Stratifolds to Exotic Spheres
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微分代数拓扑：从层层到奇异球体
DOI：
10.1090/gsm/110
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发表时间：
2010
影响因子：
0.4
通讯作者：
M. Kreck
中科院分区：
文献类型：
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作者：
M. Kreck
本书介绍了拓扑空间的同调和光滑流形的上同调的几何介绍。作者引入了一类新的分层空间，即所谓的分层空间。他从微分拓扑学中推导出基本概念，如萨德定理、单位划分和横截性。在此基础上，在层叠的框架下构造了同调群，证明了同调公理。这意味着对于好的空间，这些同调群与普通的奇异同调一致。除了利用公理进行同调群的标准计算外，还给出了重要同调类的直接构造。作者还遵循Quillen的思想定义了层叠上同调群。同样，在本说明书中，某些重要的上同调类非常自然地出现，例如，在书中构建并随后应用的特征类。最基本的结果之一，庞加莱二元性，在这种方法中几乎是微不足道的。一些基本的不变量，如欧拉特征和签名，都是从(上)同调群中得到的。这些不变量在微分拓扑学中一些最引人注目的结果中扮演着重要的角色。特别地，作者证明了Hirzebruch签名定理的一个特例，并重点介绍了Milnor奇特的7-球面。本书基于作者在美因茨和海德堡教授的课程。读者应该熟悉点集拓扑和差分拓扑的基本概念。这本书可以用来组合介绍微分和代数拓扑，以及在一门关于微分几何的课程中快速介绍(Co)同调。
This book presents a geometric introduction to the homology of topological spaces and the cohomology of smooth manifolds. The author introduces a new class of stratified spaces, so-called stratifolds. He derives basic concepts from differential topology such as Sard's theorem, partitions of unity and transversality. Based on this, homology groups are constructed in the framework of stratifolds and the homology axioms are proved. This implies that for nice spaces these homology groups agree with ordinary singular homology. Besides the standard computations of homology groups using the axioms, straightforward constructions of important homology classes are given. The author also defines stratifold cohomology groups following an idea of Quillen. Again, certain important cohomology classes occur very naturally in this description, for example, the characteristic classes which are constructed in the book and applied later on. One of the most fundamental results, Poincare duality, is almost a triviality in this approach. Some fundamental invariants, such as the Euler characteristic and the signature, are derived from (co)homology groups. These invariants play a significant role in some of the most spectacular results in differential topology. In particular, the author proves a special case of Hirzebruch's signature theorem and presents as a highlight Milnor's exotic 7-spheres. This book is based on courses the author taught in Mainz and Heidelberg. Readers should be familiar with the basic notions of point-set topology and differential topology. The book can be used for a combined introduction to differential and algebraic topology, as well as for a quick presentation of (co)homology in a course about differential geometry.