Introduction to Differential and Algebraic Topology

This introduction to topology, which is a revised, rewritten, second edition of the work first published in Russian in 1980, is a primary manual of topology. It contains the basic concepts and theorems of general topology and homotopy theory, the classification of two-dimensional surfaces, an outline of smooth manifold theory and mappings of smooth manifolds. Elements of Morse and homology theory, with their application to fixed points, are also included. Finally, the role of topology in mathematical analysis, geometry, mechanics and differential equations is illustrated. The book contains many illustrations drawn by A.T. Frenko, which, while forming an integral part of the book, also reflect the visual and philosophical aspects of modern topology. Each chapter ends with a review of the recommended literature. This text is aimed at researchers and graduate students whose work involves the application of topology, homotopy and homology theories.