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.

Preface

Ⅰ Sheaves and Presheaves

 1 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

 I 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 φ-soft and φ-fine sheaves

 10 Subspaces

 11 The Vietoris mapping theorem and homotopy invariance

 12 Relative cohomology

 13 Mayer-Vietoris theorems

 14 Continuity

 15 The Kiinneth 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

Ⅱ Comparison with Other Cohomology Theories

 1 Singular cohomology

 2 Alexander-Spanier cohomology

 3 de Rham cohomology

 4 Cech cohomology

 Exercises

Ⅳ Applications of Spectral Sequerices

 I 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 Caftan

 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-Uoore Homology

 I 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

 13 Uniqueness theorems for maps and relative homology

 14 The Kuinneth 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 Cech Homology

 I Theory of cosheaves

 2 Local triviality

 3 Local isomorphisms

 4 Cech homology

 5 The reflector

 6 Spectral sequences

 7 Coresolutions

 8 Relative Cech 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

Bibliography

List of Symbols

List of Selected Facts

Index