Algebraic Topology A First Course

Cover
Title page
Preface
PART I: CALCULUS lN THE PLANE
CHAPTER 1: Path IntegraIs
1a. DifferentiaI Forms and Path IntegraIs
lb. 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
2c. Vector Fields and Fluid Flow
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
5c. The Jordan Curve Theorem
5d. 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 Toms and Other Surfaces
8b. The Euler Characteristic
PART V: COHOMOLOGY AND HOMOLOGY, II
CHAPTER 9: Holes and IntegraIs
9a. Multiply Connected Regions
9b. Integration over Continuous Paths and Chains
9c. Periods of IntegraIs
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
llb. Lifting Paths and Homotopies
11c. G-Coverings
lld. Covering Transformations
CHAPTER 12: The Fundamental Group
l2a. Definitions and Basic Properties
l2b. Homotopy
l2c. 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
l3b. Automorphisms of Coverings
l3c. The Univers al Covering
l3d. Coverings and Subgroups of the Fundamental Group
CHAPTER 14: The Van Kampen Theorem
l4a. G-Coverings from the Universal Covering
l4b. Patching Coverings Together
l4c. The Van Kampen Theorem
l4d. Applications: Graphs and Free Groups
PART VIII: COHOMOLOGY AND HOMOLOGY , III
CHAPTER 15: Cohomology
l5a. Patching Coverings and Cech Cohomology
l5b. Cech Cohomology and Homology
l5c. De Rham Cohomology and Homology
l5d. Proof of Mayer-Vietoris for De Rham Cohomology
CHAPTER 16: Variations
l6a. The Orientation Covering
l6b. Coverings from l-Forms
l6c. Another Cohomology Group
l6d. G-Sets and Coverings
l6e. Coverings and Group Homomorphisms
l6f. G-Coverings and Cocyc1es
PART IX: TOPOLOGY OF SURFACES
CHAPTER 17: The Topology of Surfaces
l7a. Triangulation and Polygons with Sides Identified
l7b. Classification of Compact Oriented Surfaces
l7c. The Fundamental Group of a Surface
CHAPTER 18: Cohomology on Surfaces
l8a. l-Forms and Homology
l8b. Integrals of 2-Forms
l8c. Wedges and the Intersection Pairing
l8d. 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
2la. Spaces of Functions and l-Forms
2lb. Adeles
2lc. Riemann-Roch
2ld. 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
APPENDIX A: Point Set Topology
Al. Some Basic Notions in Topology
A2. Connected Components
A3. Patching
A4. Lebesgue Lemma
APPENDIX B: Analysis
BI. Results from Plane Calculus
B2. Partition of Unity
APPENDIX C: Algebra
CI. Linear Algebra
C2. Groups; Free Abelian Groups
C3. Polynomials; Gauss's Lemma
APPENDIX D: On Surfaces
Dl. Vector Fields on Plane Domains
D2. Charts and Vector Fields
D3. DifferentiaI Forms on a Surface
APPENDIX E: Proof of Borsuk's Theorem
Hints and Answers
References
Index of Symbols
Index