Lectures on Differential Topology by Riccardo Benedetti
Lectures on Differential Topology - Table of Contents
1. The Smooth Category of Open Subsets of Euclidean Spaces
2. The Category of Embedded Smooth Manifolds
3. Stiefel and Grassmann Manifolds
4. Tautological Bundles and Pull-Back
5. Compact Embedded Smooth Manifolds
6. The Category of Smooth Manifolds
7. Cut and Paste Compact Manifolds
8. Transversality
9. Morse Functions and Handle Decompositions
10. Bordism
11. Smooth Cobordism
12. Applications of Cobordism Rings
13. Line Bundles, Hypersurfaces, and Cobordism
14. Euler–Poincaré Characteristic
15. Surfaces
16. Bordism Characteristic Numbers
17. The Pontryagin–Thom Construction
18. High-Dimensional Manifolds
19. On 3-Manifolds
20. On 4-Manifolds
What You Will Learn in Lectures on Differential Topology
"Lectures on Differential Topology" by Riccardo Benedetti is a comprehensive introduction to "differential topology", designed for graduate students and researchers. It focuses on core concepts such as "smooth manifolds", "smooth maps", "transversality", the Poincaré-Hopf theorem, and cut-and-paste constructions, providing both theoretical understanding and practical, hands-on techniques. The book emphasizes intuitive geometric methods, helping readers grasp abstract ideas through concrete examples.
The text begins with the foundations of smooth manifolds, tangent spaces, and smooth maps, gradually introducing advanced topics like degree theory, characteristic numbers, bordism, cobordism, and the Pontryagin-Thom construction. It also covers the classification of compact surfaces, intersection forms, homotopy groups of spheres, and the Whitney trick. Exercises are integrated throughout, reinforcing concepts and encouraging readers to develop problem-solving skills while building geometric intuition.
Riccardo Benedetti’s approach makes this book suitable as a "textbook" for master’s and doctoral courses, or as a reference for experienced mathematicians exploring differential topology from a “bare-hands” perspective. By combining rigorous proofs with illustrative examples, it allows readers to understand and apply key tools in modern topology, from smooth manifolds and smooth maps to advanced theorems and applications, laying a solid foundation for further study in geometry and topology.
Book Details & Specifications
Title:
Lectures on Differential Topology by Riccardo Benedetti
Publisher:
American Mathematical Society
Year:
2019
Pages:
416
Type:
PDF
Language:
English
ISBN-10 #:
1470466740
ISBN-13 #:
978-1470466749
License:
arXiv License (Non-Exclusive Distribution)
Amazon:
Amazon
About the Author: Riccardo Benedetti
The author
Riccardo Benedetti
is an Italian mathematician and "professor at the University of Pisa", specializing in "topology" and geometry. He has contributed to research in low-dimensional manifolds, real algebraic geometry, and quantum topology, and has mentored many students who became active researchers in mathematics. His book, "Lectures on Differential Topology", provides a clear introduction to "smooth manifolds", "transversality", and related concepts. Designed for graduate students, it combines rigorous theory with geometric intuition and practical constructions.
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