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Full Description
Differential topology uncovers the hidden structure of smooth spaces -the foundation of modern geometry and topology. This book offers a clear, rigorous introduction to the subject, blending theory with concrete examples and applications. Beginning with the basics of manifolds and smooth maps, it develops essential tools and concepts such as tangent spaces, transversality, cobordism, and tubular neighbourhoods, before progressing to powerful invariants like the Brouwer degree, intersection numbers, and the Hopf invariant. Along the way, readers encounter landmark results including Whitney's embedding theorem, Brouwer's fixed point theorem, the Pontryagin construction, Hopf's degree theorem, and the Poincaré-Hopf index theorem. Each chapter combines intuitive explanations with precise and detailed proofs, supported by exercises and detailed solutions that deepen understanding. Ideal for advanced undergraduates, graduate students, and researchers, this text provides a gateway to one of mathematics' most elegant and influential fields - where analysis, geometry, and topology meet.
Contents
Introduction; 1. A brief introduction to topological spaces; 2. Smooth manifolds; 3. The inverse function theorem, immersions and embeddings; 4. Submersions and regular values; 5. Transversality; 6. Abstract smooth manifolds; 7. Whitney's embedding theorems; 8. Smooth homotopy; 9. Manifolds with boundary; 10. Brouwer fixed point theorem; 11. The Brouwer degree modulo 2; 12. Tubular neighbourhoods and transversality; 13. Intersection theory modulo 2; 14. Orientation; 15. The integer-valued Brouwer degree; 16. Pontryagin construction and Hopf's degree theorem; 17. Vector fields and the Poincaré-Hopf index theorem; Appendix. Solutions to selected exercises; References; Index.
