ELEMENTS OF DIFFERENTIAL TOPOLOGY

• 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
The Adjoint Action
Existence of Lie Subgroups
Foliation
• Hints/Solutions to Select Exercises
• Bibliography
• Index

Exercises appear at the end of each chapter.

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.

Features
• Provides a full review of the necessary calculus background
• Offers simple proofs of the classification of one-dimensional manifolds and the classification of compact surfaces
• Covers all elementary concepts and standard results in differential topology without assuming any prior knowledge of algebraic topology
• Includes plenty of end-of-chapter exercises, with hints and solutions at the back of the book

Author(s) Bio
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.