Algebraic Topology: A First Course

✍ Scribed by William Fulton

• Publisher: Springer-Verlag
• Year: 1995
• Tongue: English
• Leaves: 449
• Series: Series: Graduate Texts in Mathematics, Vol. 153
• Category: Library

✦ Synopsis

This book introduces the important ideas of algebraic topology by emphasizing the relation of these ideas with other areas of mathematics. Rather than choosing one point of view of modern topology (homotropy theory, axiomatic homology, or differential topology, say) the author concentrates on concrete problems in spaces with a few dimensions, introducing only as much algebraic machinery as necessary for the problems encountered. This makes it possible to see a wider variety of important features in the subject than is common in introductory texts; it is also in harmony with the historical development of the subject. The book is aimed at students who do not necessarily intend on specializing in algebraic topology.

Content Level » Lower undergraduate

Related subjects » Geometry & Topology

Cover

S Title

Algebraic Topology: A First Course

Copyright

 © 1995 Springer-Verlag
 ISBN 0-387-94327-7 (softcover)
 QA612F85 1995 514' 2—dc20
 ISBN 0-387-94327-7
 ISBN 3-540-94327-7
 SPIN 10762531

Dedication

Preface

Contents

PART I: CALCULUS IN THE PLANE

 CHAPTER 1: Path Integrals

      1a. Differential Forms and Path Integrals

      1b. When Are Path Integrals Independent of Path?

      1c. A Criterion for Exactness

 CHAPTER 2: Angles and Deformations

      2a. Angle Functions and Winding Numbers

      2b. Reparametrizing and Deforming Paths

PART II: WINDING NUMBERS

 CHAPTER 3: The Winding Number

      3a. Definition of the Winding Number

      3b. Homotopy and Reparametrization

      3c. Varying the Point

      3d. Degrees and Local Degrees

 CHAPTER 4: Applications of Winding Numbers

      4a. The Fundamental Theorem of Algebra

      4b. Fixed Points and Retractions

      4c. Antipodes

      4d. Sandwiches

PART III: COHOMOLOGY AND HOMOLOGY, I

 CHAPTER 5: De Rham Cohomology and the Jordan Curve Theorem

      5a. Definitions of the De Rham Groups

      5b. The Coboundary Map

      Sc. The Jordan Curve Theorem

      3d. Applications and Variations

 CHAPTER 6: Homology

      6a. Chains, Cycles, and H0U

      6b. Boundaries, H1U, and Winding Numbers

      6c. Chains on Grids

      6d. Maps and Homology

      6e. The First Homology Group for General Spaces

PART IV: VECTOR FIELDS

 CHAPTER 7: Indices of Vector Fields

      7a. Vector Fields in the Plane

      7b. Changing Coordinates

      7c. Vector Fields on a Sphere

 CHAPTER 8: Vector Fields on Surfaces

      8a. Vector Fields on a Torus and Other Surfaces

      8b. The Euler Characteristic

PART V: COHOMOLOGY AND HOMOLOGY, II

 CHAPTER 9: Holes and Integrals

      9a. Multiply Connected Regions

      9b. Integration over Continuous Paths and Chains

      9c. Periods of Integrals

      9d. Complex Integration

 CHAPTER 10: Mayer—Vietoris

      10a. The Boundary Map

      10b. Mayer—Vietoris for Homology

      10c. Variations and Applications

      10d. Mayer—Vietoris for Cohomology

PART VI: COVERING SPACES AND FUNDAMENTAL GROUPS, I

 CHAPTER 11: Covering Spaces

      11a. Definitions

      11b. Lifting Paths and Homotopies

      11c. G-Coverings

      11d. Covering Transformations

 CHAPTER 12: The Fundamental Group

      12a. Definitions and Basic Properties

      12b. Homotopy

      12c. Fundamental Group and Homology

PART VII: COVERING SPACES AND FUNDAMENTAL GROUPS, II

 CHAPTER 13: The Fundamental Group and Covering Spaces

      13a. Fundamental Group and Coverings

      13b. Automorphisms of Coverings

      13c. The Universal Covering

      13d. Coverings and Subgroups of the Fundamental Group

 CHAPTER 14: The Van Kampen Theorem

      14a. G-Coverings from the Universal Covering

      14b. Patching Coverings Together

      14c. The Van Kampen Theorem

      14d. Applications: Graphs and Free Groups

PART VIII: COHOMOLOGY AND HOMOLOGY, III

 CHAPTER 15: Cohomology

      15a. Patching Coverings and tech Cohomology

      15b. Cech Cohomology and Homology

      15c. De Rham Cohomology and Homology

      15d. Proof of Mayer—Vietoris fo rDe Rham Cohomology

 CHAPTER 16: Variations

      16a. The Orientation Covering

      16b. Coverings from 1-Forms

      16c. Another Cohomology Group

      16d. G-Sets and Coverings

      16e. Coverings and Group Homomorphisms

      16f. G-CoVerings and Cocycles

PART IX: TOPOLOGY OF SURFACES

 CHAPTER 17: The Topology of Surfaces

      17a. Triangulation and Polygons with Sides Identified

      17b. Classification of Compact Oriented Surfaces

      17c. The Fundamental Group of a Surface

 CHAPTER 18: Cohomology on Surfaces

      18a. 1-Forms and Homology

      18b. Integrals of 2-Forms

      18d. De Rham Theory on Surfaces

PART X: RIEMANN SURFACES

 CHAPTER 19: Riemann Surfaces

      19a. Riemann Surfaces and Analytic Mappings

      19b. Branched Coverings

      19c. The Riemann—Hurwitz Formula

 CHAPTER 20: Riemann Surfaces and Algebraic Curves

      20a. The Riemann Surface of an Algebraic Curve

      20b. Meromorphic Functions on a Riemann Surface

      20c. Holomorphic and Meromorphic 1-Forms

      20d. Riemann's Bilinear Relations and the Jacobian

      20e. Elliptic and Hyperelliptic Curves

 CHAPTER 21: The Riemann—Roch Theorem

      21a. Spaces of Functions and 1-Forms

      21b. Adeles

      21c. Riemann—Roch

      21d. The Abel—Jacobi Theorem

PART XI: HIGHER DIMENSIONS

 CHAPTER 22: Toward Higher Dimensions

      22a. Holes and Forms in 3-Space

      22b. Knots

      22c. Higher Homotopy Groups

      22d. Higher De Rham Cohomology

      22e. Cohomology with Compact Supports

 CHAPTER 23: Higher Homology

      23a. Homology Groups

      23b. Mayer—Vietoris for Homology

      23c. Spheres and Degree

      23d. Generalized Jordan Curve Theorem

 CHAPTER 24: Duality

      24a. Two Lemmas from Homological Algebra

      24b. Homology and De Rham Cohomology

      24c. Cohomology and Cohomology with Compact Supports

      24d. Simplicial Complexes

APPENDICES

 Conventions and Notation

 APPENDIX A: Point Set Topology

      A1. Some Basic Notions in Topology

      A2. Connected Components

      A3. Patching

      A4. Lebesgue Lemma

 APPENDIX B: Analysis

      B1. Results from Plane Calculus

      B2. Partition of Unity

 APPENDIX C: Algebra

      C1. Linear Algebra

      C2. Groups; Free Abelian Groups

      C3. Polynomials; Gauss's Lemma

 APPENDIX D: On Surfaces

      D1. Vector Fields on Plane Domains

      D2. Charts and Vector Fields

      D3. Differential Forms on a Surface

 APPENDIX E: Proof of Borsuk's Theorem

Hints and Answers

References

Index of Symbols

Index

✦ Table of Contents

Cover......Page 1
S Title......Page 2
Algebraic Topology: A First Course......Page 4
SPIN 10762531......Page 5
Dedication......Page 6
Preface......Page 8
Contents......Page 14
PART I: CALCULUS IN THE PLANE......Page 20
1a. Differential Forms and Path Integrals......Page 22
1b. When Are Path Integrals Independent of Path?......Page 26
1c. A Criterion for Exactness......Page 29
2a. Angle Functions and Winding Numbers......Page 36
2b. Reparametrizing and Deforming Paths......Page 42
PART II: WINDING NUMBERS......Page 52
3a. Definition of the Winding Number......Page 54
3b. Homotopy and Reparametrization......Page 57
3c. Varying the Point......Page 61
3d. Degrees and Local Degrees......Page 62
4a. The Fundamental Theorem of Algebra......Page 67
4b. Fixed Points and Retractions......Page 68
4c. Antipodes......Page 72
4d. Sandwiches......Page 75
PART III: COHOMOLOGY AND HOMOLOGY, I......Page 78
5a. Definitions of the De Rham Groups......Page 82
5b. The Coboundary Map......Page 84
Sc. The Jordan Curve Theorem......Page 87
3d. Applications and Variations......Page 91
6a. Chains, Cycles, and H0U......Page 97
6b. Boundaries, H1U, and Winding Numbers......Page 101
6c. Chains on Grids......Page 104
6d. Maps and Homology......Page 108
6e. The First Homology Group for General Spaces......Page 110
PART IV: VECTOR FIELDS......Page 114
7a. Vector Fields in the Plane......Page 116
7b. Changing Coordinates......Page 120
7c. Vector Fields on a Sphere......Page 121
8a. Vector Fields on a Torus and Other Surfaces......Page 125
8b. The Euler Characteristic......Page 132
PART V: COHOMOLOGY AND HOMOLOGY, II......Page 140
9a. Multiply Connected Regions......Page 142
9b. Integration over Continuous Paths and Chains......Page 146
9c. Periods of Integrals......Page 149
9d. Complex Integration......Page 150
10a. The Boundary Map......Page 156
10b. Mayer—Vietoris for Homology......Page 159
10c. Variations and Applications......Page 163
10d. Mayer—Vietoris for Cohomology......Page 166
PART VI: COVERING SPACES AND FUNDAMENTAL GROUPS, I......Page 170
11a. Definitions......Page 172
11b. Lifting Paths and Homotopies......Page 175
11c. G-Coverings......Page 177
11d. Covering Transformations......Page 182
12a. Definitions and Basic Properties......Page 184
12b. Homotopy......Page 189
12c. Fundamental Group and Homology......Page 192
PART VII: COVERING SPACES AND FUNDAMENTAL GROUPS, II......Page 196
13a. Fundamental Group and Coverings......Page 198
13b. Automorphisms of Coverings......Page 201
13c. The Universal Covering......Page 205
13d. Coverings and Subgroups of the Fundamental Group......Page 208
14a. G-Coverings from the Universal Covering......Page 212
14b. Patching Coverings Together......Page 215
14c. The Van Kampen Theorem......Page 216
14d. Applications: Graphs and Free Groups......Page 220
PART VIII: COHOMOLOGY AND HOMOLOGY, III......Page 224
15a. Patching Coverings and tech Cohomology......Page 226
15b. Cech Cohomology and Homology......Page 229
15c. De Rham Cohomology and Homology......Page 232
15d. Proof of Mayer—Vietoris fo rDe Rham Cohomology......Page 236
16a. The Orientation Covering......Page 238
16b. Coverings from 1-Forms......Page 239
16c. Another Cohomology Group......Page 241
16d. G-Sets and Coverings......Page 244
16e. Coverings and Group Homomorphisms......Page 246
16f. G-CoVerings and Cocycles......Page 247
PART IX: TOPOLOGY OF SURFACES......Page 250
17a. Triangulation and Polygons with Sides Identified......Page 252
17b. Classification of Compact Oriented Surfaces......Page 255
17c. The Fundamental Group of a Surface......Page 261
18a. 1-Forms and Homology......Page 266
18b. Integrals of 2-Forms......Page 270
18d. De Rham Theory on Surfaces......Page 275
PART X: RIEMANN SURFACES......Page 280
19a. Riemann Surfaces and Analytic Mappings......Page 282
19b. Branched Coverings......Page 287
19c. The Riemann—Hurwitz Formula......Page 291
20a. The Riemann Surface of an Algebraic Curve......Page 296
20b. Meromorphic Functions on a Riemann Surface......Page 300
20c. Holomorphic and Meromorphic 1-Forms......Page 303
20d. Riemann's Bilinear Relations and the Jacobian......Page 308
20e. Elliptic and Hyperelliptic Curves......Page 310
21a. Spaces of Functions and 1-Forms......Page 314
21b. Adeles......Page 318
21c. Riemann—Roch......Page 322
21d. The Abel—Jacobi Theorem......Page 325
PART XI: HIGHER DIMENSIONS......Page 332
22a. Holes and Forms in 3-Space......Page 336
22b. Knots......Page 339
22c. Higher Homotopy Groups......Page 343
22d. Higher De Rham Cohomology......Page 344
22e. Cohomology with Compact Supports......Page 347
23a. Homology Groups......Page 351
23b. Mayer—Vietoris for Homology......Page 353
23c. Spheres and Degree......Page 358
23d. Generalized Jordan Curve Theorem......Page 362
24a. Two Lemmas from Homological Algebra......Page 365
24b. Homology and De Rham Cohomology......Page 369
24c. Cohomology and Cohomology with Compact Supports......Page 374
24d. Simplicial Complexes......Page 378
Conventions and Notation......Page 384
A1. Some Basic Notions in Topology......Page 386
A2. Connected Components......Page 388
A3. Patching......Page 389
A4. Lebesgue Lemma......Page 390
B1. Results from Plane Calculus......Page 392
B2. Partition of Unity......Page 394
C1. Linear Algebra......Page 397
C2. Groups; Free Abelian Groups......Page 399
C3. Polynomials; Gauss's Lemma......Page 404
D1. Vector Fields on Plane Domains......Page 406
D2. Charts and Vector Fields......Page 408
D3. Differential Forms on a Surface......Page 410
APPENDIX E: Proof of Borsuk's Theorem......Page 412
Hints and Answers......Page 416
References......Page 438
Index of Symbols......Page 440
Index......Page 444