Sheaf Theory

This book is primarily concerned with the study of cohomology theories of general topological spaces with "general coefficient systems." Sheaves play several roles in this study. For example, they provide a suitable notion of "general coefficient systems." Moreover, they furnish us with a common method of defining various cohomology theories and of comparison between different cohomology theories. The parts of the theory of sheaves covered here are those areas important to algebraic topology. Sheaf theory is also important in other fields of mathematics, notably algebraic geometry, but that is outside the scope of the present book. Thus a more descriptive title for this book might have been Algebraic Topology from the Point of View of Sheaf Theory. Several innovations will be found in this book. Notably, the concept of the "tautness" of a subspace (an adaptation of an analogous notion of Spanier to sheaf-theoretic cohomology) is introduced and exploited throughout the book. The factthat sheaf-theoretic cohomology satisfies 1 the homotopy property is proved for general topological spaces. Also, relative cohomology is introduced into sheaf theory. Concerning relative cohomology, it should be noted that sheaf-theoretic cohomology is usually considered as a "single space" theory.

Inhalt
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.