Sheaves in Topology

verfasst von: Alexandru Dimca

Verlag: Springer Berlin Heidelberg

Buchreihe : Universitext

Enthalten in: Professional Book Archive

Einloggen, um Zugang zu erhalten

Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter

1. Derived Categories

Abstract

In the first section we recall the simplest facts of homological algebra in an abelian category A. The second section introduces the triangulated categories as a generalization of the homotopical categories K*(A). The main aim of this chapter is to introduce the derived categories and the derived functors, and this is done in the third section. The last section is devoted to a key example, the derived functor of Hom.

Alexandru Dimca

2. Derived Categories in Topology

Abstract

The first section contains various basic facts on sheaves, including the definition of (hyper) cohomology, some standard associated spectral sequences and several versions of the celebrated de Rham Theorem. After briefly discussing the derived tensor product in the second section, we give an ample introduction to the direct and inverse images of sheaves under continuous mappings in section 3. The adjunction triangle is singled out in the forth section, since this is one of the recurrent tools used in these notes. The last section is devoted to the first properties of the local systems. These are the building blocks for more complicated sheaves and, in the same time, the sheaves were the marriage between algebra and topology is easily seen.

Alexandru Dimca

3. Poincaré-Verdier Duality

Abstract

The cohomological dimension for rings and topological spaces is introduced in the first section. The second section contains the main results of Verdier duality, including the properties of the dualizing sheaf ω_X. The very general results of this section are specialized in the next section, yielding the usual Poincaré duality and Alexander duality. Here the dual sheaf complex is also introduced, and its compatibility with direct and inverse images is clearly stated. The last section contains a number of basic vanishing results for the cohomology of Stein spaces with local system coefficients.

Alexandru Dimca

4. Constructible Sheaves, Vanishing Cycles and Characteristic Varieties

Abstract

Constructible sheaves and constructible functions, objects in which algebra and topology are blended in subtle way, are introduced in the first section. The study of the variation in the topology of the fibers of a function is encoded in two functors, the nearby cycles and the vanishing cycles, whose definitions and main properties are given in the second section. The characteristic variety and the characteristic cycle, geometric ways to measure how far a constructible sheaf is from a local system, are introduced in the third section.

Alexandru Dimca

5. Perverse Sheaves

Abstract

For X a complex algebraic variety, the derived category D _c ^b (X) can be obtained starting from two natural, but quite different, abelian categories, namely the category C(X) of constructible sheaves on X and the category Perv(X) of perverse sheaves on X. The optimal way to understand this reality is the formalism of t-structures, to be introduced in the first section. The second section is devoted to the main properties of perverse sheaves and to a detailed description of germs of such sheaves in dimensions 0 and 1. The third section is a trip into the realm of D-module theory, trying to describe the dictionary behind the famous Riemann-Hilbert correspondence. Intersection cohomology, one of the sources of the perverse sheaves (maybe even their birth-place), is briefly discussed in the final section.

Alexandru Dimca

6. Applications to the Geometry of Singular Spaces

Abstract

In the first section we study hypersurface singularities and their associated objects such as Milnor fibers, links and monodromy zeta-functions. In the second section we pass to a semi-global setting, that of deformations of complex analytic varieties over a small disc. In the next section we glue this semiglobal information into a global picture in the study of a polynomial function f:ℂⁿ⁺¹→ℂ. The final section is devoted to vanishing results for the local system coefficient cohomology of hypersurface (or hyperplane) arrangement complements in a projective space ℙⁿ.

Alexandru Dimca

Backmatter

Titel: Sheaves in Topology
verfasst von: Alexandru Dimca
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-642-18868-8
Print ISBN: 978-3-540-20665-1
DOI: https://doi.org/10.1007/978-3-642-18868-8

Springer Professional

Über dieses Buch

Inhaltsverzeichnis

Frontmatter

1. Derived Categories

2. Derived Categories in Topology

3. Poincaré-Verdier Duality

4. Constructible Sheaves, Vanishing Cycles and Characteristic Varieties

5. Perverse Sheaves

6. Applications to the Geometry of Singular Spaces

Backmatter