I Sheaves and Presheaves. - Definitions. - 2 Homomorphisms subsheaves and quotient sheaves. - 3 Direct and inverse images. - 4 Cohomomorphisms. - 5 Algebraic constructions. - 6 Supports. - 7 Classical cohomology theories. - Exercises. - II Sheaf Cohomology. - 1 Differential sheaves and resolutions. - 2 The canonical resolution and sheaf cohomology. - 3 Injective sheaves. - 4 Acyclic sheaves. - 5 Flabby sheaves. - 6 Connected sequences of functors. - 7 Axioms for cohomology and the cup product. - 8 Maps of spaces. - 9 ?-soft and ?-fine sheaves. - 10 Subspaces. - 11 The Vietoris mapping theorem and homotopy invariance. - 12 Relative cohomology. - 13 Mayer-Vietoris theorems. - 14 Continuity. - 15 The Künneth and universal coefficient theorems. - 16 Dimension. - 17 Local connectivity. - 18 Change of supports; local cohomology groups. - 19 The transfer homomorphism and the Smith sequences. - 20 Steenrod's cyclic reduced powers. - 21 The Steenrod operations. - Exercises. - III Comparison with Other Cohomology Theories. -1 Singular cohomology. - 2 Alexander-Spanier cohomology. - 3 de Rham cohomology. - 4 ?ech cohomology. - Exercises. - IV Applications of Spectral Sequences. - 1 The spectral sequence of a differential sheaf. - 2 The fundamental theorems of sheaves. - 3 Direct image relative to a support family. - 4 The Leray sheaf. - 5 Extension of a support family by a family on the base space. - 6 The Leray spectral sequence of a map. - 7 Fiber bundles. - 8 Dimension. - 9 The spectral sequences of Borel and Cartan. - 10 Characteristic classes. - 11 The spectral sequence of a filtered differential sheaf. - 12 The Fary spectral sequence. - 13 Sphere bundles with singularities. - 14 The Oliver transfer and the Conner conjecture. - Exercises. - V Borel-Moore Homology. - 1 Cosheaves. - 2 The dual of a differential cosheaf. - 3 Homology theory. - 4 Maps of spaces. - 5 Subspaces and relative homology. - 6 The Vietoris theorem homotopy and covering spaces. - 7 The homology sheaf of a map. - 8 The basic spectral sequences. - 9 Poincaré duality. - 10 The cap product. - 11 Intersection theory. - 12 Uniqueness theorems. - 31 Uniqueness theorems for maps and relative homology. - 14 The Künneth formula. - 15 Change of rings. - 16 Generalized manifolds. - 17 Locally homogeneous spaces. - 18 Homological fibrations and p-adic transformation groups. - 19 The transfer homomorphism in homology. - 20 Smith theory in homology. - Exercises. - VI Cosheaves and ?ech Homology. - 1 Theory of cosheaves. - 2 Local triviality. - 3 Local isomorphisms. - 4 Cech homology. - 5 The reflector. - 6 Spectral sequences. - 7 Coresolutions. - 8 Relative ?ech homology. - 9 Locally paracompact spaces. - 10 Borel-Moore homology. - 11 Modified Borel-Moore homology. - 12 Singular homology. - 13 Acyclic coverings. - 14 Applications to maps. - Exercises. - A Spectral Sequences. - 1 The spectral sequence of a filtered complex. - 2 Double complexes. - 3 Products. - 4 Homomorphisms. - B Solutions to Selected Exercises. - Solutions for Chapter I. - Solutions for Chapter II. - Solutions for Chapter III. - Solutions for Chapter IV. - Solutions for Chapter V. - Solutions for Chapter VI. - List of Symbols. - List of Selected Facts. Language: English