An Introduction to Intersection Homology Theory, second edition
This book represents the second edition of the book of the same title but only one author, namely F. Kirwan. It appeared as volume 187 in the Pitman Research Notes in Mathematical Sciences (Longman Scientific & Technical, Harlow) in 1988. For this second edition another author was asked to cooperate with F. Kirwan and the text has been substantially extended. The original idea of the first author was to explain intersection homology and provide the reader with the proof (or at least a sketch of the proof) of the Kazhdan-Lusztig conjecture. The book in the present form can serve as a relatively easy introduction to intersection homology, showing its relations to other homology theories and presenting various applications. Not many prerequisites are required.
In the introduction the reader will find information about the homology and cohomology of manifolds and is instructed that these nice results (e.g. Poincaré duality) are no more valid if we admit manifolds with singularities. It is explained from the very beginning that if we want to extend the relevant results to manifolds with singularities then we must use intersection homology. The next two chapters still have an introductory character. We find here more details about classical simplicial and singular homology and cohomology, the theory of sheaves and sheaf cohomology, and the theory of derived categories. The fourth chapter brings a relatively elementary definition of intersection homology. Many of its basic properties are studied here. The next chapter is then devoted to the application of intersection homology to special topological pseudomanifolds called Witt spaces. Then there is a chapter on the relation between intersection cohomology and L2-cohomology, and a chapter on sheaf-theoretic intersection homology (this interpretation allows one to prove the topological invariance of intersection homology).
After a chapter devoted to perverse sheaves, there is a chapter where intersection cohomology is applied to toric varieties associated with fans. Chapter 10 is devoted to the Weil conjectures and is oriented towards the Weil conjectures for singular varieties. The next chapter introduces D-modules and the Riemann-Hilbert correspondence. Having this material available, the authors then pass on, in the last chapter, to the Kazhdan-Lusztig conjecture. The authors declare that their aim was not to write a fundamental treatise on intersection homology but rather to give propaganda for this new homology. Therefore they give, quite systematically after each chapter, recommendations for further reading. The book can be used as a first reading on intersection homology and its applications. The authors present many examples (and exercises) so that the presentation has quite a concrete character.