Topology of Singular Spaces and Constructible Sheaves [Paperback]

This volume is based on the lecture notes of six courses delivered at a Cimpa Summer School in Temuco, Chile, in January 2001. Leading experts contribute with introductory articles covering a broad area in probability and its applications, such as mathematical physics and mathematics of finance. Written at graduate level, the lectures touch the latest advances on each subject, ranging from classical probability theory to modern developments. Thus the book will appeal to students, teachers and researchers working in probability theory or related fields.

Assuming that the reader is familiar with sheaf theory, the book gives a self-contained introduction to the theory of constructible sheaves related to many kinds of singular spaces, such as cell complexes, triangulated spaces, semialgebraic and subanalytic sets, complex algebraic or analytic sets, stratified spaces, and quotient spaces. The relation to the underlying geometrical ideas are worked out in detail, together with many applications to the topology of such spaces. All chapters have their own detailed introduction, containing the main results and definitions, illustrated in simple terms by a number of examples. The technical details of the proof are postponed to later sections, since these are not needed for the applications.

1 Thom-Sebastiani Theorem for constructible sheaves.- 1.1 Milnor fibration.- 1.1.1 Cohomological version of a Milnor fibration.- 1.1.2 Examples.- 1.2 Thom-Sebastiani Theorem.- 1.2.1 Preliminaries and Thom-Sebastiani for additive functions.- 1.2.2 Thom-Sebastiani Theorem for sheaves.- 1.3 The Thom-Sebastiani Isomorphism in the derived category.- 1.4 Appendix: K?nneth formula.- 2 Constructible sheaves in geometric categories.- 2.0.1 The basic results.- 2.0.2 Definable spaces.- 2.1 Geometric categories.- 2.2 Constructible sheaves.- 2.3 Constructible functions.- 3 Localization results for equivariant constructible sheaves.- 3.1 Equivariant sheaves.- 3.1.1 ElӜ