層論

《層論(第2版)(英文版)》是由世界圖書出版公司出版的。

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

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

This book is primarily concerned with the study of cohomology theories ofgeneral topological spaces with“general coefficient systems.’’Sheaves playseveral roles in this study.For example.they provide a suitable notion of“general coefl~cient systems.”Moreover.they furnish US with a commonmethod of defining various cohomology theories and of comparison betweendifferent cohomology theories.

The parts of the theory of sheaves covered here are those areas impor.tant to algebraic topology.Sheaf theory is also important in other fields ofmathematics，notably algebraic geometry,but that iS outside the scope ofthe present book.Thus a more descriptive title for this book might havebeen Algebraic Topology b-om the Point View ol Sheaf Theory.

Several innovations will be found in this book. Notably,the con.cept of the“gautness’’of a subspace ran adaptation of an analogous no.tion of Spanier to sheaf-theoretic cohomologyl iS introduced and exploitedthroughout the book.The fact that sheaf-theoretic cohomology satisfiesthe homotopy property is proved for general topological spaces.1 Also，relative cohomology iS introduced into sheaf theory.Concerning relativecohomology,it should be noted that sheaf-theoretic cohomology iS usuallyconsidered as a“single space”theory.This is not without reason.sincecohomology relative to a closed subspace can be obtained by taking coef.ficients in a certain type of sheaf，while that relative to an open subspace（or，more generally，to a taut subspace）can be obtained by taking coho-mology with respect to a special family of supports.However，even in thesecases.it iS sometimes of notational advantage to have a relative cohomologytheory.For example，in our treatment of characteristic classes in ChapterIV the use of relative cohomology enables US to develop the theory in fullgenerality and with relatively simple notation.Our definition of relativecohomology in sheaf theory is the first fully satisfactory one to be given.It is of interest to note that.unlike absolute cohomology,the relative CO-homology groups are not the derived functors of the relative cohomologygroup in degree zero（but they usually are SO in most cases of interest）.