Texts and Readings in Mathematics

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. It will be useful to general students of pure mathematics and can be used to design a course at the M.Sc. level in Indian universities. The second part, consisting of chapters 5 to 8, caters to researchers in the areas of topology, differential or algebraic geometry and global analysis, and touches on advanced topics of general interest in these areas. Finally, the third part is meant for those who want to work 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 generalized Poincaré conjecture, and 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.
1.Basic concepts of manifolds
2.Approximation theorems and Whitney's embedding
3.Linear structures on manifolds
4.Riemannian manifolds
5.Vector bundles on manifolds
6.Transversality
7.Tubular neighbourhoods
8.Spaces of smooth maps
9.Morse theory
10.Theory of handle presentations
11.Homotopy classification of regular sections
Bibliography
Index