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Complements of Discriminants of Smooth Maps: Topology and Applications: Revised Edition
V. A. Vassiliev, Independent University of Moscow, Russia
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Translations of Mathematical Monographs
1992; 265 pp; softcover
Volume: 98
Reprint/Revision History:
revised 1994
ISBN-10: 0-8218-4618-3
ISBN-13: 978-0-8218-4618-6
List Price: US$93 Member Price: US$74.40
Order Code: MMONO/98

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.

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

Reviews

"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

• Introduction
• Cohomology of braid groups and configuration spaces
• Applications: Complexity of algorithms, superpositions of algebraic functions and interpolation theory
• Topology of spaces of real functions without complicated singularities
• Stable cohomology of complements of discriminants and caustics of isolated singularities of holomorphic functions
• Cohomology of the space of knots
• Invariants of ornaments
• Appendix $$1.$$ Classifying spaces and universal bundles. Join
• Appendix $$2.$$ Hopf algebras and $$H$$-spaces
• Appendix $$3.$$ Loop spaces
• Appendix $$4.$$ Germs, jets, and transversality theorems
• Appendix $$5.$$ Homology of local systems
• Bibliography