Aimed at those who have an elementary knowledge of linear algebra, general topology, multivariate, calculus, analysis, and algebraic topology, this book gives an introduction to some fundamental tools of differential topology. The first part comprising chapters 1 to 4 is foundational. The second part, consisting of chapters 5 to 8, caters to researchers in the areas of Topology, Geometry and Global Analysis, and touches on advanced topics of general interest in these areas. Finally, the third part is meant for those desirous of working in the field of differential topology itself. Some of the highlights of the book are Thom transversality, Morse theory, Theory of handle presentation, h-cobordism theorem and generalised Poincare conjecture, and the Gromov theory of homotopy principle of certain partial differential relations. The intention is to acquaint the reader with some epochal discoveries in the field of manifolds, mainly the earlier works of Stephen Smale for which he was awarded the fields Medal.
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