An introduction to manifolds
Manifolds, the higher-dimensional analogs of smooth curves and surfaces, are essential mathematical objects with applications in diverse fields such as classical mechanics, general relativity, and quantum field theory. By combining algebra, topology, and analysis, manifolds provide a framework for studying the geometric and topological properties of spaces. They enable the formulation of physical laws and equations, allowing physicists to analyze the dynamics of mechanical systems, describe the curvature of spacetime in general relativity, and investigate the behavior of elementary particles in quantum field theory. Manifolds serve as a unifying concept that connects abstract mathematics with the empirical observations of the physical world.
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, 'Introduction to Manifolds' is also an excellent foundation for Springer's GTM 82, 'Differential Forms in Algebraic Topology'.