Descuento:
-5%Antes:
Despues:
57,95 €Chapter 1. Prelude: Love, hate, and exponentials16
Chapter 2. Paths and homotopies30
Chapter 3. The winding number42
Chapter 4. Topology of the plane64
Chapter 5. Integrals and the winding number88
Chapter 6. Vector fields and the rotation number116
Chapter 7. The winding number in functional analysis136
Chapter 8. Coverings and the fundamental group154
Chapter 9. Coda: The Bott periodicity theorem184
Appendix A. Linear algebra196
Appendix B. Metric spaces218
Appendix C. Extension and approximation theorems232
Appendix D. Measure zero238
Appendix E. Calculus on normed spaces244
Appendix F. Hilbert space254
Appendix G. Groups and graphs264
Bibliography276
Index
The winding number is one of the most basic invariants in topology. It measures the number of times a moving point PP goes around a fixed point QQ, provided that PP travels on a path that never goes through QQ and that the final position of PP is the same as its starting position. This simple idea has far-reaching applications. The reader of this book will learn how the winding number can
• Help us show that every polynomial equation has a root (the fundamental theorem of algebra),
• Guarantee a fair division of three objects in space by a single planar cut (the ham sandwich theorem),
• Explain why every simple closed curve has an inside and an outside (the Jordan curve theorem),
• Relate calculus to curvature and the singularities of vector fields (the Hopf index theorem),
• Allow one to subtract infinity from infinity and get a finite answer (Toeplitz operators),
• Generalize to give a fundamental and beautiful insight into the topology of matrix groups (the Bott periodicity theorem).
All these subjects and more are developed starting only from mathematics that is common in final-year undergraduate courses.
Author
John Roe: Pennsylvania State University, State College, PA.
-5%
-5%
-5%
-5%
-5%
-5%
-5%
-5%
-5%
-5%
