An Introduction to Intersection Homology Theory

This book is the second edition of the book of the same title but with 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 the second author was asked to assist and the text has been substantially extended. The original idea of the first author was to explain intersection homology and bring the reader to the proof (or at least a sketch of the proof) of the Kazhdan-Lusztig conjecture. The book in the present form serves as a relatively easy introduction to intersection homology, shows its relations to other homology theories and presents various applications. Very few prerequisites are needed.

In the introduction the reader will find information about the homology and cohomology of manifolds and is instructed that many nice results (e.g. Poincaré duality) are no longer valid if we admit manifolds with singularities. From the very beginning it is stated that if we want to extend relevant results to manifolds with singularities we must use intersection homology. The next two chapters still have an introductory character. We find 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 provides a relatively elementary definition of intersection homology. Many of its basic properties are studied. 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 of intersection cohomology and L2-cohomology, and a chapter on sheaf-theoretic intersection homology (this interpretation allows us to prove the topological invariance of intersection homology). After a chapter devoted to perverse sheaves, the next chapter deals with applications of intersection cohomology to toric varieties associated with fans.

Chapter 10 is devoted to the Weil conjectures; it is oriented towards the Weil conjectures for singular varieties. The next chapter introduces D-modules and the Riemann-Hilbert correspondence. Having this material at hand, the authors progress onto the Kazhdan-Lusztig conjecture in the last chapter. The authors declare that their aim was not to write a fundamental treatise on intersection homology but rather to provide propaganda for this new homology. Therefore they give, quite systematically after each chapter, recommendations for further reading. The book will be very useful as a first reading on intersection homology and its applications. The authors present many examples (and exercises) so that the presentation has a concrete character.

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