An Introduction to Intersection Homology Theory

Now more that a quarter of a century old, intersection homology theory has proven to be a powerful tool in the study of the topology of singular spaces, with deep links to many other areas of mathematics, including combinatorics, differential equations, group representations, and number theory. Like its predecessor, An Introduction to Intersection Homology Theory, Second Edition introduces the power and beauty of intersection homology, explaining the main ideas and omitting, or merely sketching, the difficult proofs. It treats both the basics of the subject and a wide range of applications, providing lucid overviews of highly technical areas that make the subject accessible and prepare readers for more advanced work in the area. This second edition contains entirely new chapters introducing the theory of Witt spaces, perverse sheaves, and the combinatorial intersection cohomology of fans. Intersection homology is a large and growing subject that touches on many aspects of topology, geometry, and algebra. With its clear explanations of the main ideas, this book builds the confidence needed to tackle more specialist, technical texts and provides a framework within which to place them.

Table of Contents:
INTRODUCTION Poincare duality Morse theory for siningular spac es de Rham cohomology and L2-c -cohomol ology The cohomology of pr projective vari ties REVIEW OF HOMOLOGY AND COHOMOLOGY Simplicial homology Singular homology Homology with close closed support Conclusion Further reading REVIEW OF SHEAF COHOMOLOGY AND DERIVED CATEGORIES Sheaves Cech cohomology of sheaves Hypercohomology Functors and exactness Resolution of sheaves of complexes Cohomology and hypercohomology via derived functors Derived categories. Right derived functors Further reading. THE DEFINITION OF INTERSECTION HOMOLOGY Stratified spaces and pseudomanifolds Simplicial intersection homology Singular intersection homology Simple examples of intersection homology Normalisati ons Relative groups and the Mayer-Vietoris sequence. The intersection homology of a cone Functoriality of intersection homology Homology with local coefficients Quasi-projec tive complex varieties Further reading WITT SPACES AND DUALITY Generalised Poincare duality. Witt spaces Signatures of Witt spaces The Witt-bordism groups Further reading L2- COHOMOLOGY AND INTERSECTION ON COHOMOLOGY L2-cohomology and Hodge theory The L2-cohomology of a punctured cone Varieties with isolated conical singularities Locally symmetric varieties Further reading. SHEAF-THEORETIC INTERSECTION HOMOLOGY Sheaves of singular chains Constructibility and an axiomatic characterisation The topological invariance of intersection homology Duality in the derived category Further reading PERVERSE SHEAVES Perverse sheaves Perverse sheaves on varieties Nearby and vanishing cycles The decomposition theorem Further reading THE INTERSECTION COHOMOLOGY OF FANS Affine toric varieties Toric varieties from fans Maps and torus actions Projective toric varieties and convex polytopes Stratifications of toric varieties Subdivisions and desingularisations Equivariant intersection cohomology The intersection cohomology of fans Stanley's conjectures Further reading CHARACTERISTIC p AND THE WEIL CONJECTURES Statement of the Weil conjectures Basic properties of ,-adic cohomology Etale topology and cohomology The Weil conjectures for singular varieties Further reading D-MODULES AND THE RIEMANN-HILBERT CORRESPONDENCE The Riemann-Hilbert problem Differential systems over Cn Dx-modules and intersection homology The characteristic variety of a Dx-module Holonomic differential systems Examples of characteristic varieties Left and right Dx-modules Restriction of Dx-modules Regular singularities The Riemann-Hilbert correspondence Further reading THE KAZHDAN-LUSZTIG CONJECTURE Verma modules D-modules over flag manifolds Characteristic p Hecke algebras and the Kazhdan-Lusztig polynomials Further reading Bibliography Index