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Poincaré-Verdier duality and Fourier-Sato transformation

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Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 292)

Summary

In this chapter we introduce two fundamental tools for the study of sheaves on manifolds. First, following Verdier [1], we construct a right adjoint f ! to the functor Rf !. If f: YX is a continuous map of locally compact spaces satisfying suitable conditions, and if F (resp. G) belongs to D +(A X ) (resp. D +(A Y )), one gets the formula:
$$Hom(R{f_!}G,F) = Hom(G,{f^!}F)$$
(3.0.1)
.

This generalizes the classical Poincaré duality. In fact if X is reduced to a single point, if F = A and if Y is a topological manifold, one calculates f ! A (called “the dualizing complex” on Y), and shows it is isomorphic to the orientation sheaf on Y, shifted by the dimension. With the functor f ! in hands, it is then possible to derive many nice new formulas of sheaf theory.

Secondly, we make a detailed study of the Fourier-Sato transformation, an operation which interchanges conic sheaves (in the derived category) on a vector bundle and conic sheaves on the dual vector bundle.

In the course of this chapter we study sheaves on topological manifolds: orientation sheaf, flabby dimension, cohomologically constructible sheaves, and we also introduce the γ-topology, as a preparation to Chapter V.

For another approach to the contents of §1–§4 one may also consult Borel et al [1], Gelfand-Manin [1], Iversen [1] and [SHS].

Much of the material of this chapter may be considered as classical, though it had not received a systematic treatment beforehand.

Keywords

Exact Sequence Vector Bundle Open Subset Commutative Diagram Compact Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1990

Authors and Affiliations

  1. 1.Research Institute for Mathematical SciencesKyoto UniversityKyoto 606Japan
  2. 2.Department of MathematicsUniversity of Paris VIParis Cedex 05France

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