Sheaves in topology
type: Item
instance of: scholarly article
title: Sheaves in topology
zbMATH Open document ID: 1043.14003
author: Alexandru Dimca
published in: Universitext
publication date: 2004-03-23T00:00:00Z
review text: The text under review grew out of a series of lectures given by the author at the University of Bordeaux, France, during the 2000/2001 academic year. The author's aim is to provide the reader with a systematic introduction to the theory of constructible sheaves in modern algebraic topology and its various applications to singular complex varieties, and that in a comprehensible, concise and user-friendly way. Constructible sheaves, in their abstract categorical setting, appeared in algebraic topology at an early stage, first as local systems of coefficients, and were then developed, in the 1970s, as an important tool in the advanced study of the topology of (singular) complex spaces. Although the vast amount of spectacular applications of abstract sheaves and their cohomology has become sheerly overwhelming, during the last decades, and in spite of the existence of a number of excellent books devoted to this subject, many topologists and geometers seem to be still hesitant to adopt this utmost powerful theory. Admittedly, there is quite a huge formalism behind the modern abstract approach to sheaf theory, ranging from derived categories to perverse sheaves, but the recent developments in algebraic topology, algebraic geometry, complex analytic geometry, micro-local analysis, and mathematical physics can barely be followed up without a profound knowledge of it, on the other hand. The book under review covers most of the basic notions and results in the theory of constructible sheaves on complex spaces, together with numerous concrete geometric examples and applications. In addition, and along this path, the text does the service of filling the yawning gap between the very complete and highly general monographs [such as the books by \textit{M. Kashiwara} and \textit{P. Shapira}, ``Sheaves on manifolds'' (Berlin 1990; Zbl 0709.18001), \textit{A. A. Beilinson}, \textit{J. Bernstein} and \textit{P. Deligne}, ``Faisceaux pervers'', Astérisque 100 (1982; Zbl 0536.14011) or …
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 14F43
Mathematics Subject Classification ID: 14F17
Mathematics Subject Classification ID: 14B05
Mathematics Subject Classification ID: 32C18
Mathematics Subject Classification ID: 14-02
Mathematics Subject Classification ID: 14F10
Mathematics Subject Classification ID: 55R65
zbMATH DE Number: 2061739
zbMATH Keywords: sheaves
zbMATH Keywords: constructible sheaves
zbMATH Keywords: perverse sheaves
zbMATH Keywords: sheaf cohomology
zbMATH Keywords: derived categories
zbMATH Keywords: triangulated categories
zbMATH Keywords: intersection homology
zbMATH Keywords: singularities
zbMATH Keywords: \(D\)-modules
zbMATH Keywords: Poincaré-Verdier duality
zbMATH Keywords: vanishing theorems
zbMATH Keywords: intersection cohomology
zbMATH Keywords: hypersurface singularities
reviewed by: Werner Kleinert
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2061739