Description
Elements of Differential Topology
Author: Shastri Anant R.
Language: EnglishSubject for Elements of Differential Topology:
Keywords
Smooth Manifolds; Open Subset; smooth; Smooth Map; map; Vector Bundle; open; Tangent Space; subset; Tubular Neighborhood; manifolds; Lie Algebra; tangent; Oriented Manifold; space; Quotient Map; inclusion; Quotient Space; function; Inverse Function Theorem; linear; Morse Function; Lie Subgroup; Lie Group; Topological Group; Ambient Isotopic; Vector Valued Functions; Closed Subgroup; Borsuk Ulam Theorem; Connected Sum; Local Homomorphism; Normal Bundle; Nondegenerate Critical Point; Implicit Function Theorem; Lie Homomorphism
· 17.8x25.4 cm · Hardback
Description
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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. Integral Calculus. Submanifolds of Euclidean Spaces. Integration on Manifolds. Abstract Manifolds. Isotopy. Intersection Theory. Geometry of Manifolds. Lie Groups and Lie Algebras: The Basics. Hints/Solutions to Select Exercises. Bibliography. Index.
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.
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