An Introduction to Intersection Homology Theory, Second Edition

Huge advances in intersection homology theory along with significant applications of the theory have emerged in diverse areas of mathematics. Introduction to Intersection Homology Theory provides an introduction to the subject that is easily accessible to graduate students. This second edition features a chapter on intersection homology of toric varieties and the link with the theory of convex polytopes. It also includes material on Zucker's conjecture and recent progress. In addition, it discusses the historical background to the development of homology theory. Based on a course taught by the author, this textbook remains essential for graduate students in this area.

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.

INTRODUCTIONPoincaré dualityMorse theory for siningular spac esde Rham cohomology and L2-c -cohomol ology The cohomology of pr projective vari ties REVIEW OF HOMOLOGY AND COHOMOLOGY Simplicial homologySingular homology Homology with close closed supportConclusion Further reading REVIEW OF SHEAF COHOMOLOGY AND DERIVED CATEGORIES Sheaves Cech cohomology of sheavesHypercohomologyFunctors and exactnessResolution of sheaves of complexes Cohomology and hypercohomology via derived functors Derived categories. Right derived functorsFurther reading. THE DEFINITION OF INTERSECTION HOMOLOGY Stratified spaces and pseudomanifolds Simplicial intersection homologySingular intersection homologySimple examples of intersection homologyNormalisati ons Relative groups and the Mayer-Vietoris sequence. The intersection homology of a cone Functoriality of intersection homologyHomology with local coefficientsQuasi-projec tive complex varieties Further readingWITT SPACES AND DUALITY Generalised Poincaré duality. Witt spacesSignatures of Witt spacesThe Witt-bordism groups Further readingL2- COHOMOLOGY AND INTERSECTION ON COHOMOLOGY L2-cohomology and Hodge theory The L2-cohomology of a punctured cone Varieties with isolated conical singularities Locally symmetric varietiesFurther reading. SHEAF-THEORETIC INTERSECTION HOMOLOGYSheaves of singular chainsConstructibility and an axiomatic characterisationThe topological invariance of intersection homologyDuality in the derived categoryFurther readingPERVERSE SHEAVESPerverse sheavesPerverse sheaves on varietiesNearby and vanishing cyclesThe decomposition theoremFurther readingTHE INTERSECTION COHOMOLOGY OF FANSAffine toric varietiesToric varieties from fansMaps and torus actionsProjective toric varieties and convex polytopesStratifications of toric varietiesSubdivisions and desingularisationsEquivariant intersection cohomologyThe intersection cohomology of fansStanley's conjecturesFurther readingCHARACTERISTIC p AND THE WEIL CONJECTURESStatement of the Weil conjecturesBasic properties of ,-adic cohomologyÉtale topology and cohomologyThe Weil conjectures for singular varietiesFurther readingD-MODULES AND THE RIEMANN-HILBERT CORRESPONDENCEThe Riemann-Hilbert problemDifferential systems over CnDx-modules and intersection homologyThe characteristic variety of a Dx-moduleHolonomic differential systemsExamples of characteristic varietiesLeft and right Dx-modulesRestriction of Dx-modulesRegular singularitiesThe Riemann-Hilbert correspondenceFurther readingTHE KAZHDAN-LUSZTIG CONJECTUREVerma modulesD-modules over flag manifoldsCharacteristic pHecke algebras and the Kazhdan-Lusztig polynomialsFurther readingBibliographyIndex