دانلود کتاب Singularities of Mappings: The Local Behaviour of Smooth and Complex Analytic Mappings (Grundlehren der mathematischen Wissenschaften)

فهرست مطالب :

Preface
Contents
1 Introduction
1.1 Real or Complex?
1.2 Structure of the Book
1.3 The Nearby Stable Object
1.4 Exercises and Open Questions
1.5 Notation
Part I Thom-Mather Theory: Right-Left Equivalence, Stability, Versal Unfoldings, Finite Determinacy
2 Manifolds and Smooth Mappings
2.1 Germs
2.2 Manifolds and Their Tangent Spaces
Exercises for Sect.2.2
2.3 Inverse Mapping Theorem and Consequences
Exercises for Sect.2.3
2.4 Submanifolds
Exercises for Sect.2.4
2.5 Vector Fields and Flows
Exercises for Sect.2.5
2.6 Transversality
Exercises for Sect.2.6
2.7 Local Conical Structure
3 Left-Right Equivalence and Stability
3.1 Classification of Functions by Right Equivalence
Exercises for Sect.3.1
3.2 Left-Right Equivalence and Stability
3.2.1 Right Equivalence and Left Equivalence
Exercises for Sect.3.2
3.3 First Calculations
Exercises for Sect.3.3
3.4 Multi-Germs
3.4.1 Notation
Exercises for Sect.3.4
3.5 Infinitesimal Stability Implies Stability
Exercises for Sect.3.5
3.6 Stability of Multi-Germs
Exercises for Sect.3.6
4 Contact Equivalence
4.1 The Contact Tangent Space
4.2 Using TKef to Calculate TAef
Exercises for Sect.4.2
4.3 Construction of Stable Germs as Unfoldings
Exercises for Sect.4.3
4.4 Contact Equivalence
Exercises for Sect.4.4
4.5 Geometric Criterion for Finite Ae-Codimension
4.5.1 Sheafification
Exercises for Sect.4.5
4.6 Transversality
4.7 Thom–Boardman Singularities
Exercises for Sect.4.7
5 Versal Unfoldings
5.1 Versality
Exercises for Sect.5.1
5.2 Global Stability of C∞ Mappings
5.2.1 Stable Maps Are Not Always Dense
5.2.2 Mather\'s Nice Dimensions
5.3 Topological Stability
5.4 Bifurcation Sets
Exercises for Sect.5.4
5.5 The Notion of Stable Perturbation of a Map-Germ
6 Finite Determinacy
6.1 Proof of the Finite Determinacy Theorem
Exercises for Sect.6.1
6.2 Estimates for the Determinacy Degree
6.3 Determinacy and Unipotency
6.3.1 Unipotent Affine Algebraic Groups
6.3.2 Unipotent Groups of k-Jets of Diffeomorphisms
6.3.3 When Is a Closed Affine Space of Germs Contained in a G-Orbit?
6.3.4 Complexification and Determinacy Degrees
6.3.5 Notes
6.4 Complete Transversals
Exercises for Sect.6.4
6.5 Notes and Further Developments
7 Classification of Stable Germs by Their Local Algebras
7.1 Stable Germs Are Classified by Their Local Algebras
Exercises for Sect.7.1
7.2 Construction of Stable Germs as Unfoldings
Exercises for Sect.7.2
7.3 The Isosingular Locus
7.3.1 Weighted Homogeneity and Local Quasihomogeneity
7.4 Quasihomogeneity and the Nice Dimensions
7.4.1 Multi-Germs
7.4.2 The Case n≥p
7.4.3 The Case n The Set Described Is Contained in r2(n,p)
The Set Described Is All of 2r(n,p)
A1: Classification of Jets of Type 1
A2: Classification of Jets of Type 2
B: Classification of Jets of Type 3
C and D: Germs of Corank ≥4
Exercises for Sect.7.4
Part II Images and Discriminants: The Topology of Stable Perturbations
8 Stable Images and Discriminants
8.1 Introduction
8.1.1 Complex Not Real
Exercises for Sect.8.1
8.2 Review of the Milnor Fibre
8.3 The Homotopy Type of the Discriminant of a Stable Perturbation: Discriminant and Image Milnor Numbers
Exercises for Sect.8.3
8.4 Finding T1Aef in the Geometry of f: Maps from n-Space to n+1-Space
8.4.1 The Conductor Ideal
Exercises for Sect.8.4
8.5 Finding T1Aef in the Geometry of f: Sections of Stable Discriminants and Images
8.5.1 Critical Space and Discriminant
Exercises for Sect.8.5
8.6 Bifurcation Sets
8.7 Calculating the Discriminant Milnor Number
Exercises for Sect.8.7
8.8 Image Milnor Number and Ae-Codimension
8.9 Further Developments
8.9.1 Almost Free Divisors
8.9.2 Thom Polynomial Techniques
8.9.3 Does μ Constant Imply Topological Triviality?
8.9.4 The Milnor–Tjurina Relation
8.9.5 Augmentation and Concatenation: New Germs from Old
9 Multiple Points
9.1 Introduction
9.2 Choosing the Right Definition
9.2.1 Semi-Simplicial Spaces
9.2.2 When Is Dkcl(f) Reduced?
9.2.3 Irritating Notation, Occasionally Necessary
9.2.4 Equations or Procedures?
9.3 Expected Dimension
9.4 Equations for D2(f)
Exercises for Sect.9.4
9.5 Equations for Dk(f) When f Is a Corank 1 Germ
9.5.1 Generalities on Functions of One Variable
9.5.2 Application to Multiple Points
Exercises for Sect.9.5
9.6 Bifurcation Sets for Germs of Corank 1
Exercises for Sect.9.6
9.7 Disentangling a Singularity: The Geometry of a Stable Perturbation
Exercises for Sect.9.7
9.8 Blowing-Up Multiple Points
9.8.1 Construction of an Ambient Space for Kk
9.8.2 Construction of Kk(f) as Subspace of Bk(X)
Exercises for Sect.9.8
9.9 What Remains To Be Done
10 Calculating the Homology of the Image
10.1 The Alternating Chain Complex
10.1.1 Motivation
Exercises for Sect.10.1
10.2 The Image Computing Spectral Sequence
10.2.1 Towards the ICSS
10.2.2 The Filtrations
10.2.3 The Spectral Sequence of a Filtered Complex
10.2.4 The Spectral Sequences Arising from the Two Filtrations on the Total Complex of the Double Complex
Exercises for Sect.10.2
10.3 Finite Simplicial Maps
10.3.1 Triangulating Dk(f)
10.3.2 ( CAltn(D•(f)), ε•#) Is a Resolution of Cn(Y)
10.4 Finite Complex Maps Are Triangulable
10.5 Other Proofs
10.6 Cohomology
Exercises for Sect.10.6
10.7 Examples and Applications of the ICSS
10.7.1 The Reidemeister Moves
10.7.2 Reidemeister I
10.7.3 Reidemeister II
10.7.4 Reidemeister III
10.7.5 Map-Germs of Multiplicity 2
10.7.6 Codimension 1 Corank 1 Germs
10.7.7 Generalised Mayer–Vietoris
10.7.8 Relation Between AH* and H*
10.7.9 Exercises for Sect.10.7
10.8 Open Questions
11 Multiple Points in the Target: The Case of Parameterised Hypersurfaces
11.1 Finding a Presentation
11.1.1 Using Macaulay2 to Find a Presentation
Exercises for Sect.11.1
11.2 Fitting Ideals and Multiple Points in the Target
11.2.1 Are the Fitting Ideal Spaces Mk(f) Cohen–Macaulay?
Exercises for Sect.11.2
11.3 Double Points in the Target
Exercises for Sect.11.3
11.4 Ae-Codimension and Image Milnor Number of Map-Germs(Cn,S)→(Cn+1,0)
Exercises for Sect.11.4
11.5 The Rank Condition
11.6 Corank 1 Mappings: Cyclic Extensions
11.7 Duality and Symmetric Presentations
11.7.1 Gorenstein Rings and Symmetric Presentations
11.7.2 Geometrical Interpretation of the Trace Homomorphism
Exercises for Sect.11.7
11.8 Triple Points in the Target
Exercises for Sect.11.8
A Jet Spaces and Jet Bundles
Exercises for Appendix A
B Stratifications
B.1 Stratification of Sets
Exercises for Sect.B.1
B.2 Stratification of Mappings
Exercises for Sect.B.2
B.3 Semialgebraic Sets
Exercises for Sect.B.3
C Background in Commutative Algebra
C.1 Spaces and Functions on Spaces
Exercises for Sect.C.1
C.2 Associated Primes
Exercises for Sect.C.2
C.3 Dimension, Depth and Cohen–Macaulay Modules
C.3.1 Krull Dimension
C.3.2 Slicing Dimension
C.3.3 Hilbert–Samuel Dimension
C.3.4 Weierstrass Dimension
C.3.5 The Hauptidealsatz
C.3.6 Depth and Cohen–Macaulay Modules
Exercises for Sect.C.3
C.4 Free Resolutions
C.4.1 Cohen–Macaulay Modules and Freeness
C.4.2 Examples of Cohen–Macaulay Spaces
Exercises for Sect.C.4
C.5 Pulling Back Algebraic Structures
Exercises for Sect.C.5
C.6 Samuel Multiplicity
Exercises for Sect.C.6
D Local Analytic Geometry
D.1 The Preparation Theorem
Exercises for Sect.D.1
D.2 Local Properties of Analytic Sets and Finite Mappings
Exercises for Sect.D.2
D.3 Degree and Multiplicity
Exercises for Sect.D.3
D.4 Normalisation of Analytic Set-Germs
D.4.1 Extension Theorems
D.4.2 Normalisation
Exercises for Sect.D.4
E Sheaves
E.1 Presheaves and Sheaves
Exercises for Sect.E.1
E.2 Coherence
Exercises for Sect.E.2
E.3 Conservation of Multiplicity
E.3.1 Representatives
Exercises for Sect.E.3
E.4 Conservation of Multiplicity II
Exercises for Sect.E.4
References
Index