Beginner s Course in Topology

This book is the result of reworking part of a rather lengthy course of lectures of which we delivered several versions at the Leningrad and Moscow Universities. In these lectures we presented an introduction to the fundamental topics of topology: homology theory, homotopy theory, theory of bundles, and topology of manifolds. The structure of the course was well determined by the guiding term elementary topology, whose main significance resides in the fact that it made us use a rather simple apparatus. tn this book we have retained {hose sections of the course where algebra plays a subordinate role. We plan to publish the more algebraic part of the lectures as a separate book. Reprocessing the lectures to produce the book resulted in the profits and losses inherent in such a situation: the rigour has increased to the detriment of the intuitiveness, the geometric descriptions have been replaced by formulas needing interpretations, etc. Nevertheless, it seems to us tha·t the book retains the main qualities of our lectures: their elementary, systematic, and pedagogical features. The preparation of the reader is assumed to be limi ted to the usual knowledge of set ·theory, algebra, and calculus which mathematics students should master after the first year and a half of studies. The exposition is accompanied by examples and exercises. We hope that the book can be used as a topology textbook.

Contenu
Set-Theoretical Terms and Notations Used in This Book, but not Generally Adopted.- 1 Topological Spaces.- § 1. Fundamental Concepts.- §2. Constructions.- §3. Homotopies.- 2 Cellular Spaces.- §1. Cellular Spaces and Their Topological Properties.- §2. Simplicial Spaces.- §3. Homotopy Properties of Cellular Spaces.- 3 Smooth Manifolds.- §1. Fundamental Concepts.- §2. Stiefel and Grassman Manifolds.- §3. A Digression: Three Theorems from Calculus.- §4. Embeddings. Immersions. Smoothings. Approximations.- §5. The Simplest Structure Theorems.- 4 Bundles.- §1. Bundles without Group Structure.- §2. A Digression: Topological Groups and Transformation Groups.- §3. Bundles with a Group Structure.- §4. The Classification of Steenrod Bundles.- §5. Vector Bundles.- §6. Smooth Bundles.- 5 Homotopy Groups.- §1. The General Theory.- §2. The Homotopy Groups of Spheres and of Classical Manifolds.- §3. Homotopy Groups of Cellular Spaces.- §4. Weak Homotopy Equivalence.- §5. The Whitehead Product.- §6. Continuation of the Theory of Bundles.- Glossary of Symbols.