Topology of Singular Spaces and Constructible Sheaves, Softcover reprint of the original 1st ed. 2003 Monografie Matematyczne Series, Vol. 63

Langue : Anglais

Auteur : Schürmann Jörg

Couverture de l’ouvrage Topology of Singular Spaces and Constructible Sheaves

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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.

1 Thom-Sebastiani Theorem for constructible sheaves.- 1.1 Milnor fibration.- 1.2 Thom-Sebastiani Theorem.- 1.3 The Thom-Sebastiani Isomorphism in the derived category.- 1.4 Appendix: Künneth formula.- 2 Constructible sheaves in geometric categories.- 2.1 Geometric categories.- 2.2 Constructible sheaves.- 2.3 Constructible functions.- 3 Localization results for equivariant constructible sheaves.- 3.1 Equivariant sheaves.- 3.2 Localization results for additive functions.- 3.3 Localization results for Grothendieck groups and trace formulae.- 3.4 Equivariant cohomology.- 4 Stratification theory and constructible sheaves.- 4.1 Stratification theory.- 4.2 Constructible sheaves on stratified spaces.- 4.3 Base change properties.- 5 Morse theory for constructible sheaves.- 5.1 Stratified Morse theory, part I.- 5.2 Characteristic cycles and index formulae.- 5.3 Stratified Morse theory, part II.- 5.4 Vanishing cycles.- 6 Vanishing theorems for constructible sheaves.- Introduction: Results and examples.- 6.1 Proof of the results.

A new cohomological approach to constructible sheaves on stratified spaces, which doesn't use the first isotopy lemma of Thom A self-contained approach to Morse theory for constructible sheaves, including a geometric introduction to the theory of characteristic cycles Very general vanishing and Lefschetz theorems of Artin-Grothendieck type in the complex algebraic and analytic context, which apply in particular to intersection (co)homology and perverse sheaves