Sheaves in Topology

封面
Springer Science & Business Media, 2004年3月12日 - 240页

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.

 

目录

Derived Categories
1
12 Homotopical Categories K𝓐
9
13 The Derived Categories D𝓐
13
14 The Derived Functors of Hom
20
Derived Categories in Topology
23
22 Derived Tensor Products
30
23 Direct and Inverse Images
32
24 The Adjunction Triangle
43
42 Nearby and Vanishing Cycles
102
43 Characteristic Varieties and Characteristic Cycles
111
Perverse Sheaves
125
52 Properties of Perverse Sheaves
133
53 𝓓Modules and Perverse Sheaves
143
54 Intersection Cohomology
154
Applications to the Geometry of Singular Spaces
165
62 Topology of Deformations
179

25 Local Systems
47
PoincaréVerdier Duality
59
32 The Functor 𝑓¹
62
33 Poincaré and Alexander Duality
67
34 Vanishing Results
72
Constructible Sheaves Vanishing Cycles and Characteristic Varieties
81
63 Topology of Polynomial Functions
193
64 Hyperplane and Hypersurface Arrangements
208
References
223
Index
233
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热门引用章节

第231页 - A. Varchenko, Multidimensional hypergeometric functions and representation theory of Lie algebras and quantum groups, Advanced Series in Mathematical Physics 21, World Scientific, River Edge, NJ, 1995.
第231页 - On the Mixed Hodge Structure on the Cohomology of the Milnor Fiber, Math.

作者简介 (2004)

Biography Alexandru Dimca

Alexandru Dimca obtained his PhD in 1981 from the University of Bucharest. His field of interest is the topology of algebraic varieties, singularities of spaces and maps, Hodge theory and D-modules.

Dimca has been a visiting member of the Max Planck Institute in Bonn and the Institute for Advanced Study in Princeton. He is the author of three monographs and over 60 research papers published in math journals all over the world.

Dimca has extensively taught at universities in Romania, Australia, the USA, and France, and he uses this teaching experience to convey effectively, to a wider mathematical community, the abstract and difficult ideas of algebraic topology.

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