Sheaves in Topology
Sheaves in Topology
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DOI：
10.1007/978-3-642-18868-8
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发表时间：
2004-04
期刊：
影响因子：
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通讯作者：
A. Dimca
中科院分区：
文献类型：
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作者：
A. Dimca
可构造层和反层层是将奇异空间分解成光滑流形的代数对应，这是R.Thom和H.Whitney提出的一个伟大的几何思想。它们推广了数学中普遍存在的局部系统，在这类奇异空间(主要是代数和解析复变)的拓扑中有很强的应用。这门学科的导论可以被视为现代代数拓扑学的教科书，讨论具有层(而不是常)系数的空间的上同调。前5章介绍了派生范畴、层复形的正象和逆象、Verdier对偶、可构层和倒层层、消失圈和特征圈。他们还讨论了与D-模和交上同调的关系。后面的章节将这个强大的工具应用到奇点、多项式函数和超平面排列的拓扑的研究中。一些有很好来源的基本结果并没有被证明，而只是用例子和推论来陈述和说明。通过这种方式，读者可以相当迅速地从基本理论到当前的研究问题，并通过例子和练习得到支持。
Constructible and perverse sheaves are the algebraic counterpart of the decomposition of a singular space into smooth manifolds, a great geometrical idea due to R. Thom and H. Whitney. These sheaves, generalizing the local systems that are so ubiquitous in mathematics, have powerful applications to the topology of such singular spaces (mainly algebraic and analytic complex varieties). This introduction to the subject can be regarded as a textbook on modern algebraic topology, treating the cohomology of spaces with sheaf (as opposed to constant) coefficients. The first 5 chapters introduce derived categories, direct and inverse images of sheaf complexes, Verdier duality, constructible and perverse sheaves, vanishing and characteristic cycles. They also discuss relations to D-modules and intersection cohomology. Later chapters apply this powerful tool to the study of the topology of singularities, polynomial functions and hyperplane arrangements. Some fundamental results, for which excellent sources exist, are not proved but just stated and illustrated by examples and corollaries. In this way, the reader is guided rather quickly from the basic theory to current research questions, supported in this by examples and exercises.