Complements of Discriminants of Smooth Maps: Topology and Applications

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Complements of Discriminantsof Smooth Maps: Topology and Applications

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     ©1992 by the American Mathematical Society

     ISBN 0-8218-4555-1

     ISBN-10: 0-8218-4618-3

     ISBN-13: 978-0-8218-4618-6

     QA612.76.V3713 1992 514\'.24-dc2O

     LCCN 92-8176

Table of Contents

Introduction

CHAPTER I  Cohomology of Braid Groups and Configuration Spaces

     §1. Four definitions of Artin\'s braid group

          1.1. Geometric definition

          1.2. Topological definition

          1.3. Algebraic-geometric definition

          1.4. Algebraic definition

          1.5. The stable braid group

     §2. Cohomology of braid groups with trivial coefficients

          2.1. The decomposition of the space C 1(m) .

          2.2. The rings H*(Br(m), 7G2) .

          2.3. The homomorphism of the braid group into the orthogonal group

          2.4. Hopf algebra

          2.5. Cohomology of stable braid groups.

          2.6. Cohomology of braid groups with other trivial coefficients

     §3. Homology of symmetric groups and configuration spaces

          3.1. Configuration spaces

          3.2. The Hopf algebra H, (S(oo) , 7G2

          3.3. Cellular decompositions of configuration spaces.

          3.4. Stable cellular decomposition

          3.5. Cellular realization of the group H* (S (m) , Z2)

          equals 0.3.6. The dual realization of the Hopf algebra H* (S(oo) , Z2) and the homology of configuration spaces

          3.7. Homomorphism from the symmetric group to the orthogonal group.

     §4. Cohomology of braid groups and configuration spaces with coefficients in the sheaf ±Z

          4.1. The sheaf ±Z and representation ±Z

          4.4. Cellular cohomolog

          4.5. More about the map of the braid group into the orthogonal group

     §5. Cohomology of braid groups with coefficients in the Coxeter representation

CHAPTER II  Applications: Complexity of Algorithms, and Superpositions of Algebraic Functions

     §1. The Schwarz genus

          1.1. Definition and elementary properties

          1.2. Genus and category

          1.3. The estimate of the genus using the cohomological length of f

          1.4. Calculation of the genus of a fibration in terms of global sections

          1.5. Homological genus of a principal covering.

     §2. Topological complexity of algorithms, and the genus of a fibration

          2.1. Statement of problems

          2.2. Algorithms

          2.3. Smale\'s theorem.

     §3. Estimates of the topological complexity of finding roots of polynomials in one variable

          3.1. Statement of results.

          3.2. Proof of Theorem 3.1.1

          3.3. An algorithm of complexity m - 1: proof of Theorem 3.1.2

          3.5. Proof of Theorem 3.1.4.

          3.6. Proof of Proposition 3.1.5.

     §4. Topological complexity of solving systems of equations in several variables

          4.1. Formulation of problems; spaces of systems.

          4.2. Statement of results

          4.3. Proof of the lower estimates.

          4.4. Upper estimates

          4.5. Two problems

     §5. Obstructions to representing algebraic functions by superpositions

          5.1. Algebraic functions

          5.2. Superpositions of algebraic functions

CHAPTER III  Topology of Spaces of Real Functions without Complicated Singularities

     § 1. Statements of reduction theorems

          1.1. The first main theorem

          1.2. The second main theorem

          1.5. Singularities of functions on one-dimensional manifolds

     §2. Spaces of functions on one-dimensional manifolds without zeros of multiplicity three

          2.1. Notation

          2.2. Statements

          2.3. Proof of Theorem 1.

          2.4. Proof of Theorem 2.

     §3. Cohomology of spaces of polynomials without multiple roots

          3.1. Statement of results

          3.2. Two preliminary remarks about the spaces Pd - Ek

          3.3. The spectral sequence for computing the groups H*

          3.4. Proof of Theorem 2.

          3.5. Proof of Theorem 3.

     §4. Proof of the first main theorem

          4.1. Statement of the main homological result

          4.2. Finite-dimensional approximations in the space of functions on M M.

          4.3. Proof of Theorem 4.2.1

          4.4. Proof of Theorem 4.2.3

          4.5. Proof of Theorem 4.2.4

          4.6. Proof of Theorem 4.2.2.

          4.7. Proof of Theorem 2 of 4.1

          4.8. On the proof of the homological part of Theorem 3\' of §3

     §5. Cohomology of spaces of maps from m-dimensional spaces to m-connected spaces

          5.1. Description of the term Ep

          5.3. Proof of Lemma 1.

          5.4. Spaces of maps with restrictions on subsets

          5.5. Example: loop spaces of spheres

          5.6. Degeneration of the spectral sequence for S2mS\".

          5.7. The action of the Freudenthal homomorphism on spectral sequences.

     §6. On the cohomology of complements of arrangements of planes in R\"

CHAPTER IV  Stable Cohomology of Complements of Discriminants and Caustics of Isolated Singularities of Holomorphic Functions

     §1. Singularities of holomorphic functions, their deformations and discriminants

          1.1. Deformations and discriminants.

          1.2. The Milnor bundle

          1.3. The sufficient jet theorem

          1.4. Versal deformations

          1.5. Adjacency of singularities. Stable irreducibility of singularities and multisingularities

     §2. Definition and elementary properties of the stable cohomology of complements of discriminants

          2.1. Local cohomology of complements of discriminants

          2.2. Maps of local cohomology groups given by adjacency of singularities

          2.3. Definition of stable cohomology for singularities in a fixed number of variables.

          2.4. Stabilization with respect to the number of variables

          2.5. Consequences of the definitions

     §3. Stable cohomology of complements of discriminants, and the loop spaces

          3.2. Stages (A), (B), and (E) of the program from 3.1.

          3.4. The isomorphism of the truncated spectral sequences for the cohomology of D6 - E(F) and S22\"S2\"+1 ; stage (D) of the program from 3.1.

     §4. Cohomological Milnor bundles

          4.1. Proof of Theorem 5.

          4.2. Proof of Theorem 7.

          4.3. Proof of Theorem 6.

     §5. Proofs of technical results

     §6. Stable cohomology of complements of caustics, and other generalizations

          6.2. The Smale-Hirsch principle for stable complements of strata of singularities.

     §7. Complements of resultants for pairs of polynomials in C

          7.1. Proof of assertion (A) of the theorem.

          7.2. Proof of Segal\'s theorem (part (B) of Theorem 4).

CHAPTER V  Cohomology of the Space of Knots

     0.1. Noncompact knots

     0.2. Geometric realization of the spectral sequence

     0.3. Coding and computation of invariants

     0.4. The groups El

     0.5. Agreement of basis invariants for symmetric knots

     0.6. Outline of the chapter

     §1. Definitions and notation

          1.1. Noncompact knots

          1.2. Approximation of kno

     §2. Basic spectral sequence

          2.1. Configurations of singularities and self-intersections

          2.2. Generating collections

          2.3. Resolution of a discriminant

          2.4. Filtration in the resolution of a discriminant

          2.5. Fundamental properties of the basic spectral sequence.

          2.6. The stable spectral sequence and stable invariants.

     §3. The term EI of the basic spectral sequence

          3.1. Decomposition of the space Qt - ai- I by the types of configurations.

          3.2. Cellular decomposition of spaces a, - ai-1 for stable values of i

          3.3. The integral spectral sequence

          3.4. An auxiliary filtration in the set o -

          3.5. The bundle structure in J-blocks.

          3.6. The generating complex of (A, b)-configurations.

          3.7. Proof of Theorem 3.6.3.

     §4. Algorithms for computing the invariants and their values

          4.1. Truncated spectral sequence.

          4.2. Actuality table

          4.3. Algorithm for computing the values of invariants.

          4.4. The extended actuality table

          4.5. The differential d1 of the truncated spectral sequence and completionof the (i - 1)th level of the actuality table.

          4.7. Proof of the theorem from 0.5.

     §5. The simplest invariants and their values for tabular knots

     §6. Conjectures, problems, and additional remarks

          6.1. Stabilization conjecture

APPENDIX 1  Classifying Spaces and Universal Bundles. Join

     1. Principal bundles

     2. Universal G-bundle

     3. The spaces K(G, 1).

     4. Universal 0(m)- and SO(m)-bundles.

     6. The Milnor construction of the universal G-bundle

APPENDIX 2  Hopf Algebras and H-Spaces

APPENDIX 3  Loop Spaces

APPENDIX 4  Germs, Jets, and Transversality Theorems

     1. Germs.

     2. Jets.

     3. Jet extensions

     4. Transversality

     5. Whitney topologies in function spaces

APPENDIX 5  Homology of Local Systems

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