Shastri Elements of Differential Topology

1. Auflage 2011
ISBN: 978-1-4398-3163-2
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 319 Seiten

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.

Zielgruppe

Undergraduate and graduate students and researchers in topology and geometry.

Autoren/Hrsg.

Shastri, Anant R.

Fachgebiete

• Mathematik | Informatik Mathematik Topologie
• Mathematik | Informatik Mathematik Geometrie

Inhaltsverzeichnis

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.

Über Autor(innen)

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.