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Ⅰ Sheaves and Presheaves 1

1 Definitions 1

2 Homomorphisms,subsheaves,and quotient sheaves 8

3 Direct and inverse images 12

4 Cohomomorphisms 14

5 Algebraic constructions 16

6 Supports 21

7 Classical cohomology theories 24

Exercises 30

Ⅱ Sheaf Cohomology 33

1 Differential sheaves and resolutions 34

2 The canonical resolution and sheaf cohomology 36

3 Injective sheaves 41

4 Acyclic sheaves 46

5 Flabby sheaves 47

6 Connected sequences of functors 52

7 Axioms for cohomology and the cup product 56

8 Maps of spaces 61

9 Ф-soft and Ф-fine sheaves 65

10 Subspaces 71

11 The Vietoris mapping theorem and homotopy invariance 75

12 Relative cohomology 83

13 Mayer-Vietoris theorems 94

14 Continuity 100

15 The Künneth and universal coefficient theorems 107

16 Dimension 110

17 Local connectivity 126

18 Change of supports;local cohomology groups 134

19 The transfer homomorphism and the Smith sequences 137

20 Steenrod's cyclic reduced powers 148

21 The Steenrod operations 162

Exercises 169

Ⅲ Comparison with Other Cohomology Theories 179

1 Singular cohomology 179

2 Alexander-Spanier cohomology 185

3 de Rham cohomology 187

4 ？ech cohomology 189

Exercises 194

Ⅳ Applications of Spectral Sequences 197

1 The spectral sequence of a differential sheaf 198

2 The fundamental theorems of sheaves 202

3 Direct image relative to a support family 210

4 The Leray sheaf 213

5 Extension of a support family by a family on the base space 219

6 The Leray spectral sequence of a map 221

7 Fiber bundles 227

8 Dimension 237

9 The spectral sequences of Borel and Cartan 246

10 Characteristic classes 251

11 The spectral sequence of a filtered differential sheaf 257

12 The Fary spectral sequence 262

13 Sphere bundles with singularities 264

14 The Oliver transfer and the Conner conjecture 267

Exercises 275

Ⅴ Borel-Moore Homology 279

1 Cosheaves 281

2 The dual of a differential cosheaf 289

3 Homology theory 292

4 Maps of spaces 299

5 Subspaces and relative homology 303

6 The Vietoris theorem,homotopy,and covering spaces 317

7 The homology sheaf of a map 322

8 The basic spectral sequences 324

9 Poincaré duality 329

10 The cap product 335

11 Intersection theory 344

12 Uniqueness theorems 349

13 Uniqueness theorems for maps and relative homology 358

14 The Künneth formula 364

15 Change of rings 368

16 Generalized manifolds 373

17 Locally homogeneous spaces 392

18 Homological fibrations and p-adic transformation groups 394

19 The transfer homomorphism in homology 403

20 Smith theory in homology 407

Exercises 411

Ⅵ Cosheaves and ？ech Homology 417

1 Theory of cosheaves 418

2 Local triviality 420

3 Local isomorphisms 421

4 ？ech homology 424

5 The reflector 428

6 Spectral sequences 431

7 Coresolutions 432

8 Relative ？ech homology 434

9 Locally paracompact spaces 438

10 Borel-Moore homology 439

11 Modified Borel-Moore homology 442

12 Singular homology 443

13 Acyclic coverings 445

14 Applications to maps 446

Exercises 448

A Spectral Sequences 449

1 The spectral sequence of a filtered complex 449

2 Double complexes 451

3 Products 453

4 Homomorphisms 454

B Solutions to Selected Exercises 455

Solutions for Chapter Ⅰ 455

Solutions for Chapter Ⅱ 459

Solutions for Chapter Ⅲ 472

Solutions for Chapter Ⅳ 473

Solutions for Chapter Ⅴ 480

Solutions for Chapter Ⅵ 486

Bibliography 487

List of Symbols 491

List of Selected Facts 493

Index 495