Poincaré-Verdier duality and Fourier-Sato transformation
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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 , Gelfand-Manin , Iversen  and [SHS].
Much of the material of this chapter may be considered as classical, though it had not received a systematic treatment beforehand.
KeywordsExact Sequence Vector Bundle Open Subset Commutative Diagram Compact Space
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