Complements of Discriminants of Smooth Maps: Topology and Applications

Description

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. Cover Title page Contents Introduction Chapter I. Cohomology of braid groups and configuration spaces Chapter II. Applications: Complexity of algorithms, superpositions of algebraic functions and interpolation theory Chapter III. Topology of spaces of real functions without complicated singularities Chapter IV. Stable cohomology of complements of discriminants and caustics of isolated singularities of holomorphic functions Chapter V. Cohomology of the space of knots Chapter VI. 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 Added in second edition Back Cover