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