**Translations of Mathematical Monographs**

1992;
265 pp;
Softcover

MSC: Primary 55; 57;

Print ISBN: 978-0-8218-4618-6

Product Code: MMONO/98

List Price: $98.00

AMS Member Price: $78.40

MAA Member Price: $88.20

**Electronic ISBN: 978-1-4704-4510-2
Product Code: MMONO/98.E**

List Price: $98.00

AMS Member Price: $78.40

MAA Member Price: $88.20

#### Supplemental Materials

# Complements of Discriminants of Smooth Maps: Topology and Applications: Revised Edition

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*V. A. Vassiliev*

This book studies a large class of topological spaces, many of which play an important role in differential and homotopy topology, algebraic geometry, and catastrophe theory. These include spaces of Morse and generalized Morse functions, iterated loop spaces of spheres, spaces of braid groups, and spaces of knots and links. Vassiliev develops a general method for the topological investigation of such spaces. One of the central results here is a system of knot invariants more powerful than all known polynomial knot invariants. In addition, a deep relation between topology and complexity theory is used to obtain the best known estimate for the numbers of branchings of algorithms for solving polynomial equations. In this revision, Vassiliev has added a section on the basics of the theory and classification of ornaments, information on applications of the topology of configuration spaces to interpolation theory, and a summary of recent results about finite-order knot invariants. Specialists in differential and homotopy topology and in complexity theory, as well as physicists who work with string theory and Feynman diagrams, will find this book an up-to-date reference on this exciting area of mathematics.

#### Readership

Physicists who work with string theory and Feynman diagrams, and specialists in differential and homotopy topology and in complexity theory.

#### Reviews & Endorsements

The book is a work of stunning originality and an impressive unification of very diverse strands … [it] is carefully planned and well written.

-- Zentralblatt MATH

#### Table of Contents

# Table of Contents

## Complements of Discriminants of Smooth Maps: Topology and Applications: Revised Edition

- Cover Cover11
- Title page v6
- Contents vii8
- Introduction 112
- Chapter I. Cohomology of braid groups and configuration spaces 1930
- Chapter II. Applications: Complexity of algorithms, superpositions of algebraic functions and interpolation theory 4354
- Chapter III. Topology of spaces of real functions without complicated singularities 7788
- Chapter IV. Stable cohomology of complements of discriminants and caustics of isolated singularities of holomorphic functions 123134
- Chapter V. Cohomology of the space of knots 153164
- Chapter VI. Invariants of ornaments 195206
- Appendix 1. Classifying spaces and universal bundles. Join 241252
- Appendix 2. Hopf algebras and 𝐻-spaces 245256
- Appendix 3. Loop spaces 247258
- Appendix 4. Germs, jets, and transversality theorems 249260
- Appendix 5. Homology of local systems 253264
- Bibliography 257268
- Added in second edition 264275
- Back Cover Back Cover1279