Primarily concerned with the study of cohomology theories of general topological spaces with "general coefficient systems", the parts of sheaf theory covered here are those areas important to algebraic topology. Among the many innovations in this book, the concept of the "tautness" of a subspace is introduced and exploited; the fact that sheaf theoretic cohomology satisfies the homotopy property is proved for general topological spaces; and relative cohomology is introduced into sheaf theory. A list of exercises at the end of each chapter helps students to learn the material, and solutions to many of the exercises are given in an appendix. This new edition of a classic has been substantially rewritten and now includes some 80 additional examples and further explanatory material, as well as new sections on Cech cohomology, the Oliver transfer, intersection theory, generalised manifolds, locally homogeneous spaces, homological fibrations and p- adic transformation groups. Readers should have a thorough background in elementary homological algebra and in algebraic topology.
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 Kunneth 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 Poincare duality.- 10 The cap product.- 11 Intersection theory.- 12 Uniqueness theorems.- 31 Uniqueness theorems for maps and relative homology.- 14 The Kunneth 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.
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2nd ed. 1997. Softcover reprint of the original 2nd ed. 1997