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Intersection cohomology. (Notes of a Seminar on Intersection Homology at the University of Bern, Switzerland, Spring 1983). (English) Zbl 0553.14002

Progress in Mathematics, Vol. 50. Swiss Seminars. Boston-Basel-Stuttgart: Birkhäuser. X, 234 p. DM. 58.00 (1984).
This volume consists of notes of a seminar going through basic properties of intersection cohomology. However 60% of its length is devoted to a major account by Borel of sheaf theory as it applies to intersection cohomology. The sections are as follows: 1) Introduction to piecewise linear intersection homology (p. 1-22), by A. Haefliger. - 2) From PL to sheaf theory (p. 23-34), by N. Habegger. - 3) A sample computation of intersection homology (p. 35-40), by M. Goresky and R. MacPherson. - 4) Pseudomanifold structures on complex analytic spaces (p. 41-46) (brief statement of a result of B. Teissier) (p. 209- 220), by N. A’Campo. - 5) Sheaf theoretic intersection cohomology (p. 47-182), by A. Borel (with the collaboration of N. Spaltenstein). - 6) Les foncteurs de la categorie des faisecaux associés à une application continue (p. 183-208), by P. P. Grivel. - 7) Witt space cobordism theory (after P. Siegel), by M. Goresky. - 8) Lefschetz fixed point theorems and intersection homology, by M. Goresky and R. MacPherson. - 9) Problems and bibliography on intersection homology (p. 221-228), by M. Goresky and R. MacPherson. - All of these - except the last - are brief accounts of topics treated more fully elsewhere; they are written to be independent of each other, apart from basic notions.
The long chapter 5 by A. Borel is a major contribution to the literature on sheaf theory. The traditional books in the subject (Godement, Swan, Bredon), written twenty years ago, do not include a number of topics which have been developed later. - Notable are the functors \(f_ !\), and \(f^ !\), which are explained here in chapter 6, by P. P. Grivel - a most useful account. Borel’s chapter includes a rather detailed account of the properties of these, and other functors both of sheaves and in the derived category: these are introduced as necessary in the study of intersection cohomology. The development here is modelled on the paper of M. Goresky and R. MacPherson, ”Intersection Homology. II”, Invent. Math. 72, 77-129 (1983; Zbl 0529.55007): Deligne’s iterative construction of perverse sheaves is axiomatised; the axioms are modified, and used to prove topological invariance; there is an extended discussion of duality, including a general account of Verdier duality.
Nowhere in this book the special properties of intersection cohomology of complex analytic spaces with middle perversity are treated. As these seem to be crucial to all the major applications (and to 75% of the papers in the bibliography) this is a highly significant - if understandable - omission: there seems at present to be no expository account in existence.
Reviewer: C.T.C.Wall

MSC:

14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14-06 Proceedings, conferences, collections, etc. pertaining to algebraic geometry
55-06 Proceedings, conferences, collections, etc. pertaining to algebraic topology
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
14F99 (Co)homology theory in algebraic geometry
00Bxx Conference proceedings and collections of articles

Citations:

Zbl 0529.55007