Differential Topology

This book offers a clear, rigorous introduction to differential topology – a cornerstone of modern mathematics with applications in geometry, physics, and engineering. It makes key concepts like transversality, homotopy, and cobordism accessible, building readers' intuition with illustrative examples, exercises, and solutions to enable self-study.

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.