Sheaves in Topology
Springer Berlin (Verlag)
978-3-540-20665-1 (ISBN)
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.
Biography Alexandru Dimca
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 Hom.- 2 Derived Categories in Topology.- 2.1 Generahties on Sheaves.- 2.2 Derived Tensor Products.- 2.3 Direct and Inverse Images.- 2.4 The Adjunction Triangle.- 2.5 Local Systems.- 3 Poincaré-Verdier Duality.- 3.1 Cohomological Dimension of Rings and Spaces.- 3.2 The Functor f!.- 3.3 Poincaré and Alexander Duality.- 3.4 Vanishing Results.- 4 Constructible Sheaves, Vanishing Cycles and Characteristic Varieties.- 4.1 Constructible Sheaves.- 4.2 Nearby and Vanishing Cycles.- 4.3 Characteristic Varieties and Characteristic Cycles.- 5 Perverse Sheaves.- 5.1 t-Structures and the Definition of Perverse.- 5.2 Properties of Perverse.- 5.3 D-Modules and Perverse.- 5.4 Intersection Cohomology.- 6 Applications to the Geometry of Singular Spaces.- Singularities, Milnor Fibers and Monodromy.- Topology of Deformations.- Topology of Polynomial Functions.- Hyperplane and Hypersurface Arrangements.- References.
From the reviews:
"The book ... is devoted to showing topologists and geometers what perverse sheaves are and what they are good for. ... The author uses the language of sheaves as a unifying framework, pointing out the special features in the topological and analytic cases. Throughout the book he gives examples and motivations for the different results and concepts. ... The material covered is very extensive ... ." (Ana Jeremías López, Mathematical Reviews, 2005j)
"The author's aim is to provide the reader with a systematic introduction to the theory of constructible sheaves ... . the reader will meet a number of theorems that appear, step-by-step, in increasing generality and applicability. This methodological strategy is very pleasant, in that it makes the text overall enlightening and instructive ... . Written in a very lucid, detailed and rigorous style, with an abundance of concrete examples and applications, it serves ... in a perfect way. Also experts in the field will find quite a bit of novelties ... ." (Werner Kleinert, Zentralblatt MATH, Vol. 1043 (18), 2004)
"This book covers a majority of basic notions and results in the theory of constructive sheaves on complex spaces and it provides a rich amount of geometrical examples and applications. The author takes the reader from simpler, older results to the most powerful and general results currently available. ... The book is well written and I would like to recommend it to anybody interested in the topic." (EMS Newsletter, December, 2005)
| Erscheint lt. Verlag | 12.3.2004 |
|---|---|
| Reihe/Serie | Universitext |
| Zusatzinfo | XVI, 240 p. |
| Verlagsort | Berlin |
| Sprache | englisch |
| Maße | 155 x 235 mm |
| Gewicht | 510 g |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Analysis |
| Mathematik / Informatik ► Mathematik ► Geometrie / Topologie | |
| Schlagworte | algebraic topology • cohomology • constructible • Garbe (Mathematik) • Homology • Monodromy • perverse • sheaf • Topology • variety |
| ISBN-10 | 3-540-20665-5 / 3540206655 |
| ISBN-13 | 978-3-540-20665-1 / 9783540206651 |
| Zustand | Neuware |
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