Constructible and perverse sheaves are the algebraic counterpart of the decomposition of a singular space into smooth manifolds. 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 author helps readers progress quickly from the basic theory to current research questions, thoroughly supported along the way by examples and exercises.
Biography Alexandru Dimca Alexandru Dimca obtained his PhD in 1981 from the University of Bucharest. His field of interest is the topology of algebraic varieties, singularities of spaces and maps, Hodge theory and D-modules. Dimca has been a visiting member of the Max Planck Institute in Bonn and the Institute for Advanced Study in Princeton. He is the author of three monographs and over 60 research papers published in math journals all over the world. Dimca has extensively taught at universities in Romania, Australia, the USA, and France, and he uses this teaching experience to convey effectively, to a wider mathematical community, the abstract and difficult ideas of algebraic topology.
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 Poincare-Verdier Duality.- 3.1 Cohomological Dimension of Rings and Spaces.- 3.2 The Functor f!.- 3.3 Poincare 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.