Singularities of mappings: the local behaviour of smooth and complex analytic mappings
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cham, Switzerland
Springer
[2020]
|
| Series: | Grundlehren der mathematischen Wissenschaften
357 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xv, 567 Seiten Illustrationen |
| ISBN: | 3030344398 9783030344399 |
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MARC
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| 245 | 1 | 0 | |a Singularities of mappings |b the local behaviour of smooth and complex analytic mappings |c David Mond, Juan J. Nuño-Ballesteros |
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Record in the Search Index
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| adam_text | David Mond • Juan J Nuno-Ballesteros
Singularities of Mappings
The Local Behaviour of Smooth and Complex
Analytic Mappings
Springer
Contents
1 Introduction 1
1 1 Real or Complex? 2
1 2 Structure of the Book 3
1 3 The Nearby Stable Object 6
1 4 Exercises and Open Questions 10
1 5 Notation 10
Part I Thom-Mather Theory: Right-Left Equivalence, Stability,
Versal Unfoldings, Finite Determinacy
2 Manifolds and Smooth Mappings 13
2 1 Germs 13
2 2 Manifolds and Their Tangent Spaces 15
2 3 Inverse Mapping Theorem and Consequences 24
2 4 Submanifolds 29
2 5 Vector Fields and Flows 32
2 6 Transversality 40
2 7 Local Conical Structure 44
3 Left-Right Equivalence and Stability 45
3 1 Classification of Functions by Right Equivalence 46
3 2 Left-Right Equivalence and Stability 58
321 Right Equivalence and Left Equivalence 71
3 3 First Calculations 74
3 4 Multi-Germs 81
341 Notation 81
3 5 Infinitesimal Stability Implies Stability 88
3 6 Stability of Multi-Germs 92
4 Contact Equivalence 97
4 1 The Contact Tangent Space 97
4 2 Using TJ amp;ef to Calculate Tsrfef 102
xi
Contents
xii
4 3 Construction of Stable Germs as Unfoldings 105
4 4 Contact Equivalence 108
4 5 Geometric Criterion for Finite rt^-Codimension
451 Sheafificalion U?
4 6 Transversality 122
4 7 Thom-Boardman Singularities 128
5 Versal Unfoldings 141
5 1 Versality 142
5 2 Global Stability of C00 Mappings 156
521 Stable Maps Are Not Always Dense 158
522 Mather’s Nice Dimensions 16®
5 3 Topological Stability 162
5 4 Bifurcation Sets 163
5 5 The Notion of Stable Perturbation of a Map-Germ 126
6 Finite Determinacy 181
6 1 Proof of the Finite Determinacy Theorem 184
6 2 Estimates for the Determinacy Degree 191
6 3 Determinacy and Unipotency 200
631 Unipotent Affine Algebraic Groups 204
632 Unipotent Groups of Jfc-Jets of Diffeomorphisms 206
633 When Is a Closed Affine Space of Germs
Contained in a S^-Orbit? 209
634 Complexification and Determinacy Degrees 209
635 Notes 209
6 4 Complete Transversals 210
6 5 Notes and Further Developments 215
7 Classification of Stable Germs by Their Local Algebras 217
7 1 Stable Germs Are Classified by Their Local Algebras 217
7 2 Construction of Stable Germs as Unfoldings 223
7 3 The Isosingular Locus 227
731 Weighted Homogeneity and Local Quasihomogeneity 231
7 4 Quasihomogeneity and the Nice Dimensions 232
741 Multi-Germs 235
742 The Case n p 236
743 The Case n p 238
Part II Images and Discriminants: The Topology of Stable
Perturbations
8 Stable Images and Discriminants 253
8 1 Introduction 253
811 Complex Not Real 258
8 2 Review of the Milnor Fibre 25
Contents
xiii
8 3 The Homotopy Type of the Discriminant of a Stable
Perturbation: Discriminant and Image Milnor Numbers 261
8 4 Finding T^f in the Geometry of /: Maps from n-Space
to n+ 1-Space 269
841 The Conductor Ideal 275
8 5 Finding f in the Geometry of /: Sections of Stable
Discriminants and Images 279
851 Critical Space and Discriminant 283
8 6 Bifurcation Sets 289
8 7 Calculating the Discriminant Milnor Number 292
8 8 Image Milnor Number and -Codimension 298
8 9 Further Developments 299
891 Almost Free Divisors 299
892 Thom Polynomial Techniques 300
893 Does Constant Imply Topological Triviality? 300
894 The Milnor-Tjurina Relation 300
895 Augmentation and Concatenation: New Germs
from Old 301
9 Multiple Points 303
9 1 Introduction 303
9 2 Choosing the Right Definition 304
921 Semi-Simplicial Spaces 311
922 When Is (/) Reduced? 312
923 Irritating Notation, Occasionally Necessary 312
924 Equations or Procedures? 315
9 3 Expected Dimension 315
9 4 Equations for D2(f) 320
9 5 Equations for Dk(f) When / Is a Corank 1 Germ 327
951 Generalities on Functions of One Variable 327
952 Application to Multiple Points 333
9 6 Bifurcation Sets for Germs of Corank 1 341
9 7 Disentangling a Singularity: The Geometry of a Stable
Perturbation 345
9 8 Blowing-Up Multiple Points 351
981 Construction of an Ambient Space for Kk 352
982 Construction of Kkif) as Subspace of Bk{X) 355
9 9 What Remains To Be Done 366
10 Calculating the Homology of the Image 369
10 1 The Alternating Chain Complex 370
10 1 1 Motivation 373
10 2 The Image Computing Spectral Sequence 380
10 2 1 Towards the ICSS 384
10 2 2 The Filtrations 385
XIV
Contents
10 2 3 The Spectral Sequence of a Filtered Complex 385
10 2 4 The Spectral Sequences Arising from the Two
Filtrations on the Total Complex of the Double
Complex 386
10 3 Finite Simplicial Maps 389
10 3 1 Triangulating Dk{f) 391
10 3 2 (C„A11(D*(/)),e;) Is a Resolution of C„(T) 394
10 4 Finite Complex Maps Are Triangulable 398
10 5 Other Proofs 399
10 6 Cohomology 399
10 7 Examples and Applications of the ICSS 402
10 7 1 The Reidemeisler Moves 403
10 7 2 Reidemeisler I 403
10 7 3 Reidemeister II 404
10 7 4 Reidemeister III 405
10 7 5 Map-Germs of Multiplicity 2 406
10 7 6 Codimension 1 Corank 1 Germs 409
10 7 7 Generalised Mayer-Vietoris 410
10 7 8 Relation Between AHt and H, 411
10 7 9 Exercises for Sect 10 7 411
10 8 Open Questions 412
II Multiple Points in the Target: The Case of Parameterised
Hypersurfaces 413
11 1 Finding a Presentation 414
11 1 1 Using Macaulay2 to Find a Presentation 418
11 2 Fitting Ideals and Multiple Points in the Target 421
11 2 1 Are the Fitting Ideal Spaces Mk(f)
Cohen-Macaulay? 428
11 3 Dou ble Points in the Target 431
11 4 c/f-Codimension and Image Milnor Number
of Map-Germs(C S) - (C+1,0) 436
11 5 The Rank Condition 442
11 6 Corank 1 Mappings: Cyclic Extensions 447
11 7 Duality and Symmetric Presentations 451
11 7 1 Gorenstein Rings and Symmetric Presentations 455
11 7 2 Geometrical Interpretation of the Trace
Homomorphism 458
11 8 Triple Points in the Target 462
A Jet Spaces and Jet Bundles 46
B Stratifications 47
B l Stratification of Sets 47
B 2 Stratification of Mappings 48
B 3 Semialgebraic Sets 48
Contents xv
C Background in Commutative Algebra 489
C l Spaces and Functions on Spaces 489
C 2 Associated Primes 493
C 3 Dimension, Depth and Cohen-Macaulay Modules 495
C31 Krull Dimension 495
C32 Slicing Dimension 496
C33 Hilbert-Samuel Dimension 498
C34 Weierstrass Dimension 498
C35 The Hauptidealsatz 498
C36 Depth and Cohen-Macaulay Modules 500
C 4 Free Resolutions 503
C41 Cohen-Macaulay Modules and Freeness 504
C42 Examples of Cohen-Macaulay Spaces 505
C 5 Pulling Back Algebraic Structures 508
C 6 Samuel Multiplicity 513
D Local Analytic Geometry 517
D l The Preparation Theorem 517
D 2 Local Properties of Analytic Sets and Finite Mappings 521
D 3 Degree and Multiplicity 525
D 4 Normalisation of Analytic Set-Germs 528
D41 Extension Theorems 531
D42 Normalisation 532
E Sheaves 537
E 1 Presheaves and Sheaves 537
E 2 Coherence 541
E 3 Conservation of Multiplicity 545
E31 Representatives 545
E 4 Conservation of Multiplicity II 549
References 553
|
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| bvnumber | BV046422008 |
| classification_rvk | SK 350 SK 240 |
| ctrlnum | (OCoLC)1141140594 (DE-599)KXP1689555432 |
| discipline | Mathematik |
| format | Book |
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| indexdate | 2024-07-10T08:44:11Z |
| institution | BVB |
| isbn | 3030344398 9783030344399 |
| language | English |
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| spelling | Mond, David 1950- Verfasser (DE-588)1024678164 aut Singularities of mappings the local behaviour of smooth and complex analytic mappings David Mond, Juan J. Nuño-Ballesteros Cham, Switzerland Springer [2020] © 2020 xv, 567 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Grundlehren der mathematischen Wissenschaften 357 Singularität Mathematik (DE-588)4077459-4 gnd rswk-swf Abbildung Mathematik (DE-588)4000044-8 gnd rswk-swf Singularität Mathematik (DE-588)4077459-4 s Abbildung Mathematik (DE-588)4000044-8 s DE-604 Nuño-Ballesteros, Juan J. Verfasser (DE-588)1204380287 aut Erscheint auch als Online-Ausgabe 978-3-030-34440-5 Grundlehren der mathematischen Wissenschaften 357 (DE-604)BV000000395 357 HEBIS Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=031834423&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Mond, David 1950- Nuño-Ballesteros, Juan J. Singularities of mappings the local behaviour of smooth and complex analytic mappings Grundlehren der mathematischen Wissenschaften Singularität Mathematik (DE-588)4077459-4 gnd Abbildung Mathematik (DE-588)4000044-8 gnd |
| subject_GND | (DE-588)4077459-4 (DE-588)4000044-8 |
| title | Singularities of mappings the local behaviour of smooth and complex analytic mappings |
| title_auth | Singularities of mappings the local behaviour of smooth and complex analytic mappings |
| title_exact_search | Singularities of mappings the local behaviour of smooth and complex analytic mappings |
| title_full | Singularities of mappings the local behaviour of smooth and complex analytic mappings David Mond, Juan J. Nuño-Ballesteros |
| title_fullStr | Singularities of mappings the local behaviour of smooth and complex analytic mappings David Mond, Juan J. Nuño-Ballesteros |
| title_full_unstemmed | Singularities of mappings the local behaviour of smooth and complex analytic mappings David Mond, Juan J. Nuño-Ballesteros |
| title_short | Singularities of mappings |
| title_sort | singularities of mappings the local behaviour of smooth and complex analytic mappings |
| title_sub | the local behaviour of smooth and complex analytic mappings |
| topic | Singularität Mathematik (DE-588)4077459-4 gnd Abbildung Mathematik (DE-588)4000044-8 gnd |
| topic_facet | Singularität Mathematik Abbildung Mathematik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=031834423&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000395 |
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