Sheaves in topology拓扑学中的层

2004-4  

北京燕山出版社  

Dimca, Alexandru  

236  

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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.

1 Derived Categories 1.1 Categories of Complexes C*(A) 1.2 Homotopical Categories K*(A) 1.3 The Derived Categories D*(A) 1.4 The Derived Functors of Hom2 Derived Categories in Topology 2.1 Generalities on Sheaves 2.2 Derived Tensor Products 2.3 Direct and Inverse Images 2.4 The Adjunction Triangle 2.5 Local Systems3 Poincaré-Verdier Duality 3.1 Cohomological Dimension of Rings and Spaces 3.2 The Functorf! 3.3 Poincare and Alexander Duality 3.4 Vanishing Results4 Constructible Sheaves, Vanishing Cycles and Characteristic Varieties 4.1 Constructible Sheaves 4.2 Nearby and Vanishing Cycles 4.3 Characteristic Varieties and Characteristic Cycles5 Perverse Sheaves 5.1 t-Structures and the Definition of Perverse Sheaves 5.2 Properties of Perverse Sheaves 5.3 D-Modules and Perverse Sheaves 5.4 Intersection Cohomology6 Applications to the Geometry of Singular Spaces 6.1 Singularities, Milnor Fibers and Monodromy 6.2 Topology of Deformations 6.3 Topology of Polynomial Functions 6.4 Hyperplane and Hypersurface ArrangementsReferencesIndex

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