Algebraic Topology by Fulton, William -- Read -- Imperial Library of Trantor

Index

Cover Title Page Copyright Page Preface Table of 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 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 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

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 Čech Cohomology 15b. Čech Cohomology and Homology 15c. De Rham Cohomology and Homology 15d. Proof of Mayer-Vietoris for De 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 18c. Wedges and the Intersection Pairing 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 APPENDIX A Point Set Topology APPENDIX B Analysis APPENDIX C Algebra APPENDIX D On Surfaces APPENDIX E Proof of Borsuk’s Theorem References

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