Topology of Singular Spaces and Constructible Sheaves

TLDR: In this article, the Thom-Sebastiani Theorem for constructible sheaves has been studied in geometric and complex spaces, and the results show that the construction of sheaves on such spaces is possible.

Abstract: 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: Kunneth 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.