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 Geometrie
• Mathematik | Informatik Mathematik Topologie

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.