# Introduction to Topology

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*V. A. Vassiliev*

This English translation of a Russian book presents the basic notions of differential and algebraic topology, which are indispensable for specialists and useful for research mathematicians and theoretical physicists. In particular, ideas and results are introduced related to manifolds, cell spaces, coverings and fibrations, homotopy groups, intersection index, etc. The author notes, “The lecture note origins of the book left a significant imprint on its style. It contains very few detailed proofs: I tried to give as many illustrations as possible and to show what really occurs in topology, not always explaining why it occurs.” He concludes, “As a rule, only those proofs (or sketches of proofs) that are interesting per se and have important generalizations are presented.”

#### Readership

Graduate students, research mathematicians, and theoretical physicists.

#### Reviews & Endorsements

The book will be very convenient for those who want to be acquainted with the topic in a short time.

-- European Mathematical Society Newsletter

A concise treatment of differential and algebraic topology.

-- American Mathematical Monthly

In little over 140 pages, the book goes all the way from the definition of a topological space to homology and cohomology theory, Morse theory, Poincaré theory, and more … emphasizes intuitive arguments whenever possible … a broad survey of the field. It is often useful to have an overall picture of a subject before engaging it in detail. For that, this book would be a good choice.

-- MAA Online

The book is based on a course given by the author in 1996 to first and second year students at Independent Moscow University … the emphasis is on illustrating what is happening in topology, and the proofs (or their ideas) covered are those which either have important generalizations or are useful in explaining important concepts … This is an excellent book and one can gain a great deal by reading it. The material, normally requiring several volumes, is covered in 123 pages, allowing the reader to appreciate the interaction between basic concepts of algebraic and differential topology without being buried in minutiae.

-- Mathematical Reviews

#### Table of Contents

# Table of Contents

## Introduction to Topology

- Cover Cover11 free
- Title iii4 free
- Copyright vi7 free
- Contents vii8 free
- Foreword xi12 free
- Chapter 1. Topological spaces and operations with them 116 free
- Chapter 2. Homotopy groups and homotopy equivalence 924
- Chapter 3. Coverings 2136
- Chapter 4. Cell spaces (CW-complexes) 2540
- Chapter 5. Relative homotopy groups and the exact sequence of a pair 3550
- Chapter 6. Fiber bundles 4156
- Chapter 7. Smooth manifolds 4964
- Chapter 8. The degree of a map 5974
- Chapter 9. Homology: Basic definitions and examples 6984
- Chapter 10. Main properties of singular homology groups and their computation 8398
- Chapter 11. Homology of cell spaces 95110
- Chapter 12. Morse theory 103118
- §12.1. Morse functions 103118
- §12.2. The cellular structure of a manifold endowed with a Morse function 104119
- §12.3. Attaching handles 106121
- §12.4. Regular Morse functions 106121
- §12.5. The boundary operator in a Morse complex 110125
- §12.6. Morse inequalities 114129
- §12.7. Standard bifurcations of Morse functions 115130

- Chapter 13. Cohomology and Poincaré duality 119134
- Chapter 14. Some applications of homology theory 129144
- Chapter 15. Multiplication in cohomology (and homology) 137152
- §15.1. Homology and cohomology groups of a Cartesian product 137152
- §15.2. Multiplication in cohomology 140155
- §15.3. Examples of multiplication in cohomology and its geometric meaning 142157
- §15.4. Main properties of multiplication in cohomology 143158
- §15.5. Connection with the de Rham cohomology 144159
- §15.6. Pontryagin multiplication 144159

- Index of Notations 145160 free
- Subject Index 147162
- BackCover BackCover1165