层论

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 Steenrods 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 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

本书主要讲述具有一般系数体系拓扑空间的上同调理论。层论包括对代数拓扑很重要的领域。书中有好多创新点，引进不少新概念，全书内容贯穿一致。证实了广义同调空间中层理论上同调满足同调基本特性的事实。将相对上同调引入层理论中。
　　读者有一定的基本同调代数和代数拓扑知识就可以理解本书。每章末都附有练习，这些可以帮助学生更好的理解书中的知识体系。附录给出了部分习题的解答。第二版中在内容上做了较大的改动，增加了80多例子和大量更深层次的内容，如，Cech上同调、Oliver变换、插值理论、广义流形、局部齐性空间、同调纤维和p进变换群。目次：层和准层；层上同调；与其他上同调定理的比较；谱序列的应用；Borel-Moore同调；上层和ech同调。
　　读者对象：数学专业的高年级本科生、研究生和相关专业的学者。