Ordinary homology theory for manifolds has beeen applied successfully in many different branches of mathematics. However, it fails when dealing with problems involving singular spaces. About 12 years ago, Goreski and MacPherson introduced a refinement, the intersection homology. It coincides with ordinary homology for manifolds, but works better when the space has singularities. The idea is to consider a subcomplex of the chain complex consisting of those chains which behave well in the singular points, that is intersect the singularities in a prescribed way. Since its introduction, intersection homology has made rapid progress and has proved extremely useful for other areas of mathematics, e.g., among others, the theory of differential equations, representation theory and number theory.
It is the aim of the book in question, not only to give an introduction to the basic ideas of intersection homology theory, but also to give applications, that is to describe how the theory is used to prove some theorems about differential equations and representation theory of Lie algebras. This goal is very well achieved. The book is written very clearly and should serve well not only beginners but also nonspecialists as a first introduction. It is a pleasure to read it.
As for the contents: In the first part the basics of intersection homology theory are outlined. After introducing the problem and a short review on ordinary homology theories, the definition of intersection homology groups is given together with some examples and first results. Then the relation to \(L^ 2\)-cohomology is given and finally Deligne’s construction of the intersection homology sheaf, which shows that these groups are independent of the chosen stratification. The second part deals with applications. The ultimate goal is to sketch Bernstein’s proof of the Kazhdan-Lusztig conjectures. To do this, the relations of representation theory of complex Lie algebras to the theory of Hecke algebras via \({\mathfrak D}\)-modules and intersection theory have to be described.