1st Edition

# Elements of Differential Topology

By Anant R. Shastri Copyright 2011
320 Pages 43 B/W Illustrations
by CRC Press

319 Pages
by CRC Press

Also available as eBook on:

Derived from the author’s course on the subject, Elements of Differential Topology explores the vast and elegant theories in topology developed by Morse, Thom, Smale, Whitney, Milnor, and others. It begins with differential and integral calculus, leads you through the intricacies of manifold theory, and concludes with discussions on algebraic topology, algebraic/differential geometry, and Lie groups.

The first two chapters review differential and integral calculus of several variables and present fundamental results that are used throughout the text. The next few chapters focus on smooth manifolds as submanifolds in a Euclidean space, the algebraic machinery of differential forms necessary for studying integration on manifolds, abstract smooth manifolds, and the foundation for homotopical aspects of manifolds. The author then discusses a central theme of the book: intersection theory. He also covers Morse functions and the basics of Lie groups, which provide a rich source of examples of manifolds. Exercises are included in each chapter, with solutions and hints at the back of the book.

A sound introduction to the theory of smooth manifolds, this text ensures a smooth transition from calculus-level mathematical maturity to the level required to understand abstract manifolds and topology. It contains all standard results, such as Whitney embedding theorems and the Borsuk–Ulam theorem, as well as several equivalent definitions of the Euler characteristic.

Review of Differential Calculus
Vector Valued Functions
Directional Derivatives and Total Derivative
Linearity of the Derivative
Inverse and Implicit Function Theorems
Lagrange Multiplier Method
Differentiability on Subsets of Euclidean Spaces
Richness of Smooth Maps

Integral Calculus
Multivariable Integration
Sard’s Theorem
Exterior Algebra
Differential Forms
Exterior Differentiation
Integration on Singular Chains

Submanifolds of Euclidean Spaces
Basic Notions
Manifolds with Boundary
Tangent Space
Special Types of Smooth Maps
Transversality
Homotopy and Stability

Integration on Manifolds
Orientation on Manifolds
Differential Forms on Manifolds
Integration on Manifolds
De Rham Cohomology

Abstract Manifolds
Topological Manifolds
Abstract Differentiable Manifolds
Gluing Lemma
Classification of One-Dimensional Manifolds
Tangent Space and Tangent Bundle
Tangents as Operators
Whitney Embedding Theorems

Isotopy
Normal Bundle and Tubular Neighborhoods
Orientation on Normal Bundle
Vector Fields and Isotopies
Patching-up Diffeomorphisms

Intersection Theory
Transverse Homotopy Theorem
Oriented Intersection Number
Degree of a Map
Nonoriented Case
Winding Number and Separation Theorem
Borsuk–Ulam Theorem
Hopf Degree Theorem
Lefschetz Theory
Some Applications

Geometry of Manifolds
Morse Functions
Morse Lemma
Operations on Manifolds
Further Geometry of Morse Functions
Classification of Compact Surfaces

Lie Groups and Lie Algebras: The Basics
Review of Some Matrix Theory
Topological Groups
Lie Groups
Lie Algebras
Canonical Coordinates
Topological Invariance
Closed Subgroups
Existence of Lie Subgroups
Foliation

Hints/Solutions to Select Exercises

Bibliography

Index

Exercises appear at the end of each chapter.

### Biography

Anant R. Shastri is a professor in the Department of Mathematics at the Indian Institute of Technology, Bombay. His research interests encompass topology and algebraic geometry.

… in Shastri’s treatment, the subject [differential forms] is developed in the larger context of the author’s stated goals, which makes for very good motivation and increased accessibility. Shastri does an excellent job with this foundational material. … It’s altogether a solid introduction to serious themes likely to persuade the reader to go deeper into the subject. Shastri’s exposition is rigorous at the same time that it evinces a light touch, and this of course makes for a very readable book. Examples abound, proofs are done in detail and include discussion along the lines of what one might hear in a good lecture presentation, and there are exercises replete with hints or solutions. Pedagogically, Elements of Differential Topology clearly gets very high marks. It is a good and useful textbook.
MAA Reviews, July 2011

Professor Shastri’s book gives an excellent point of entry to this fascinating area of mathematics by providing the basic motivation and background needed for the study of differential geometry, algebraic topology, and Lie groups. … A major strength of Professor Shastri’s book is that detailed arguments are given in places where other books leave too much for the reader to supply on his/her own. This, together with the large quantity of accessible exercises, makes this book particularly reader friendly as a stable text for an introductory course in differential topology.
—From the Foreword by F. Thomas Farrell, Binghamton, New York, USA