Ebook: An Introduction to Manifolds

Manifolds, the higher-dimensional analogs of smooth curves and surfaces, are fundamental objects in modern mathematics. Combining aspects of algebra, topology, and analysis, manifolds have also been applied to classical mechanics, general relativity, and quantum field theory.

In this streamlined introduction to the subject, the theory of manifolds is presented with the aim of helping the reader achieve a rapid mastery of the essential topics. By the end of the book the reader should be able to compute, at least for simple spaces, one of the most basic topological invariants of a manifold, its de Rham cohomology. Along the way the reader acquires the knowledge and skills necessary for further study of geometry and topology. The requisite point-set topology is included in an appendix of twenty pages; other appendices review facts from real analysis and linear algebra. Hints and solutions are provided to many of the exercises and problems.

This work may be used as the text for a one-semester graduate or advanced undergraduate course, as well as by students engaged in self-study. Requiring only minimal undergraduate prerequisites, An Introduction to Manifolds is also an excellent foundation for Springer GTM 82, Differential Forms in Algebraic Topology.

 

---

Manifolds, the higher-dimensional analogs of smooth curves and surfaces, are fundamental objects in modern mathematics. Combining aspects of algebra, topology, and analysis, manifolds have also been applied to classical mechanics, general relativity, and quantum field theory.

In this streamlined introduction to the subject, the theory of manifolds is presented with the aim of helping the reader achieve a rapid mastery of the essential topics. By the end of the book the reader should be able to compute, at least for simple spaces, one of the most basic topological invariants of a manifold, its de Rham cohomology. Along the way the reader acquires the knowledge and skills necessary for further study of geometry and topology. The requisite point-set topology is included in an appendix of twenty pages; other appendices review facts from real analysis and linear algebra. Hints and solutions are provided to many of the exercises and problems.

This work may be used as the text for a one-semester graduate or advanced undergraduate course, as well as by students engaged in self-study. Requiring only minimal undergraduate prerequisites, An Introduction to Manifolds is also an excellent foundation for Springer GTM 82, Differential Forms in Algebraic Topology.

 

Content:
Front Matter....Pages i-xviii
Smooth Functions on a Euclidean Space....Pages 5-10
Tangent Vectors in Rn as Derivations....Pages 11-18
Alternating k-Linear Functions....Pages 19-32
Manifolds....Pages 33-44
Smooth Maps on a Manifold....Pages 47-55
Quotients....Pages 57-62
The Tangent Space....Pages 63-74
Submanifolds....Pages 77-89
Categories and Functors....Pages 91-100
The Rank of a Smooth Map....Pages 101-104
The Tangent Bundle....Pages 105-117
Bump Functions and Partitions of Unity....Pages 119-126
Vector Fields....Pages 127-134
Lie Groups....Pages 135-146
Lie Algebras....Pages 149-160
Differential 1-Forms....Pages 161-171
Differential k-Forms....Pages 175-179
The Exterior Derivative....Pages 181-188
Orientations....Pages 189-198
Manifolds with Boundary....Pages 201-209
Integration on a Manifold....Pages 211-220
De Rham Cohomology....Pages 221-231
The Long Exact Sequence in Cohomology....Pages 235-242
The Mayer–Vietoris Sequence....Pages 243-248
Homotopy Invariance....Pages 249-255
Computation of de Rham Cohomology....Pages 257-262
Proof of Homotopy Invariance....Pages 263-271
Back Matter....Pages 273-277
....Pages 280-364