Topology of Singular Spaces and Constructible Sheaves

ISBN-13: 9783764321895 / Angielski / Twarda / 2003 / 454 str.

Jvrg Sch]rmann; Jorg Schurmann; Joe Schurmann

Opis

Assuming that the reader is familiar with sheaf theory, the book gives a self-contained introduction to the theory of constructible sheaves related to many kinds of singular spaces, such as cell complexes, triangulated spaces, semialgebraic and subanalytic sets, complex algebraic or analytic sets, stratified spaces, and quotient spaces. The relation to the underlying geometrical ideas are worked out in detail, together with many applications to the topology of such spaces. All chapters have their own detailed introduction, containing the main results and definitions, illustrated in simple terms by a number of examples. The technical details of the proof are postponed to later sections, since these are not needed for the applications.

Spis treści

1 Thom-Sebastiani Theorem for constructible sheaves.- 1.1 Milnor fibration.- 1.1.1 Cohomological version of a Milnor fibration.- 1.1.2 Examples.- 1.2 Thom-Sebastiani Theorem.- 1.2.1 Preliminaries and Thom-Sebastiani for additive functions.- 1.2.2 Thom-Sebastiani Theorem for sheaves.- 1.3 The Thom-Sebastiani Isomorphism in the derived category.- 1.4 Appendix: Künneth formula.- 2 Constructible sheaves in geometric categories.- 2.0.1 The basic results.- 2.0.2 Definable spaces.- 2.1 Geometric categories.- 2.2 Constructible sheaves.- 2.3 Constructible functions.- 3 Localization results for equivariant constructible sheaves.- 3.1 Equivariant sheaves.- 3.1.1 Equivariant sheaves and monodromic complexes.- 3.1.2 Equivariant derived categories.- 3.1.3 Examples and stalk formulae.- 3.2 Localization results for additive functions.- 3.3 Localization results for Grothendieck groups and trace formulae.- 3.3.1 Grothendieck groups.- 3.3.2 Trace formulae.- 3.4 Equivariant cohomology.- 4 Stratification theory and constructible sheaves.- 4.1 Stratification theory.- 4.1.1 A cohomological version of the first isotopy lemma.- 4.1.2 Comparison of different regularity conditions.- 4.1.3 Micro-local characterization of constructible sheaves.- 4.2 Constructible sheaves on stratified spaces.- 4.2.1 Cohomologically cone-like stratifications.- 4.2.2 Stability results for constructible sheaves.- 4.3 Base change properties.- 4.3.1 Some constructions for stratifications.- 4.3.2 Base change isomorphisms.- 5 Morse theory for constructible sheaves.- 5.0.1 Real stratified Morse theory.- 5.0.2 Complex stratified Morse theory.- 5.0.3 Introduction to characteristic cycles.- 5.1 Stratified Morse theory, part I.- 5.1.1 Local Morse data.- 5.1.2 Normal Morse data.- 5.1.3 Morse theory for a stratified space with corners.- 5.2 Characteristic cycles and index formulae.- 5.2.1 Index formulae and Euler obstruction.- 5.2.2 A specialization argument.- 5.3 Stratified Morse theory, part II.- 5.3.1 Normal Morse data are independent of choices.- 5.3.2 Splitting of the local Morse data.- 5.3.3 Normal Morse data and micro-localization.- 5.4 Vanishing cycles.- 6 Vanishing theorems for constructible sheaves.- Introduction: Results and examples.- 6.0.1 (Co)stalk properties.- 6.0.2 Intersection (co)homology and perverse sheaves.- 6.0.3 Vanishing results in the complex context.- 6.0.4 Nearby and vanishing cycles.- 6.0.5 Artin-Grothendieck type theorems.- 6.0.6 Applications to constructible functions.- 6.1 Proof of the results.