Singularities of mappings : the local behaviour of smooth and complex analytic mappings / David Mond, Juan J.Nuño-Ballesteros.

Format:

Book

Author/Creator:

Mond, D. (David), author.

Nuño-Ballesteros, Juan J., author.

Series:

Grundlehren der mathematischen Wissenschaften ; 357.

Grundleheren der mathematischen Wissenschaften ; 357.

Language:

English

Subjects (All):

Singularities (Mathematics).

Mappings (Mathematics).

Physical Description:

xv, 567 pages : illustrations ; 24 cm.

Place of Publication:

Cham, Switzerland : Springer, [2020]

Contents:

Introduction

Real or Complex?

Structure of the Book

The Nearby Stable Object

Exercises and Open Questions

Notation

Thom-Mather Theory : Right-Left Equivalence, Stability, Versal Unfoldings, Finite Determinacy

Manifolds and Smooth Mappings

Germs

Manifolds and Their Tangent Spaces

Inverse Mapping Theorem and Consequences

Submanifolds

Vector Fields and Flows

Transversality

Local Conical Structure

Left-Right Equivalence and Stability

Classification of Functions by Right Equivalence

Right Equivalence and Left Equivalence

First Calculations

Multi-Germs

Infinitesimal Stability Implies Stability

Stability of Multi-Germs

Contact Equivalence

The Contact Tangent Space

Using T Kef to Calculate T Aef

Construction of Stable Germs as Unfoldings

Contact Equivalence Geometric Criterion for Finite Ae-Codimension

Sheafification

Thom-Boardman Singularities

Versal Unfoldings

Versality

Global Stability of C Mappings

Stable Maps Are Not Always Dense

Mather's Nice Dimensions

Topological Stability

Bifurcation Sets

The Notion of Stable Perturbation of a Map-Germ

Finite Determinacy

Proof of the Finite Determinacy Theorem

Estimates for the Determinacy Degree

Determinacy and Unipotency

Unipotent Affine Algebraic Groups

Unipotent Groups of k-Jets of Diffeomorphisms

When Is a Closed Affine Space of Germs Contained in a G-Orbit?

Complexification and Determinacy Degrees

Notes

Complete Transversals

Notes and Further Developments

Classification of Stable Germs by Their Local Algebras

Stable Germs Are Classified by Their Local Algebras

The Isosingular Locus Weighted Homogeneity and Local Quasihomogeneity

Quasihomogeneity and the Nice Dimensions

The Case n >̲ p

The Case n < p

Images and Discriminants : The Topology of Stable Perturbations

Stable Images and Discriminants

Introduction

Complex Not Real

Review of the Milnor Fibre

The Homotopy Type of the Discriminant of a Stable Perturbation : Discriminant and Image Milnor Numbers

Finding T1Aef in the Geometry of f : Maps from n-Space to n + 1-Space

The Conductor Ideal

Finding T1Aef in the Geometry of f : Sections of Stable Discriminants and Images

Critical Space and Discriminant

Calculating the Discriminant Milnor Number

Image Milnor Number and Ae-Codimension

Further Developments

Almost Free Divisors

Thorn Polynomial Techniques

Does ... Constant Imply Topological Triviality?

The Milnor-Tjurina Relation^^^

The Isosingular Locus Augmentation and Concatenation : New Germs from Old

Multiple Points

Choosing the Right Definition

Semi-Simplicial Spaces

When Is D...(f) Reduced?

Irritating Notation, Occasionally Necessary

Equations or Procedures?

Expected Dimension

Equations for D2(f)

Equations for Dk(f) When d Is a Corank 1 Germ

Generalities on Functions of One Variable

Application to Multiple Points

Bifurcation Sets for Germs of Corank 1

Disentangling a Singularity : The Geometry of a Stable Perturbation

Blowing-Up Multiple Points

Construction of an Ambient Space for Kk

Construction of Kk(f) as Subspace of Bk(X)

What Remains To Be Done

Calculating the Homology of the Image

The Alternating Chain Complex

Motivation

The Image Computing Spectral Sequence

Towards the ICSS

The Filtrations

The Spectral Sequence of a Filtered Complex

The Spectral Sequences Arising from the Two Filtrations on the Total Complex of the Double Complex

Finite Simplicial Maps

Triangulating Dk(f)

(C...) Is a Resolution of Cn(Y)

Finite Complex Maps Are Triangulable

Other Proofs

Cohomology

Examples and Applications of the ICSS

The Reidemeister Moves

Reidemeister I

Reidemeister II

Reidemeister III

Map-Germs of Multiplicity 2

Codimension 1 Corank 1 Germs

Generalised Mayer-Vietoris

Relation Between AH* and H*

Exercises for Sect. 10.7

Open Questions

Multiple Points in the Target : The Case of Parameterised Hypersurfaces

Finding a Presentation

Using Macaulay2 to Find a Presentation

Fitting Ideals and Multiple Points in the Target

Are the Fitting Ideal Spaces Mk(f) Cohen-Macaulay? Double Points in the Target

Ae-Codimension and Image Milnor Number of Map-Germs ...

The Rank Condition

Corank 1 Mappings : Cyclic Extensions

Duality and Symmetric Presentations

Gorenstein Rings and Symmetric Presentations

Geometrical Interpretation of the Trace Homomorphism

Triple Points in the Target

Jet Spaces and Jet Bundles

Stratifications

Stratification of Sets

Stratification of Mappings

Semialgebraic Sets

Background in Commutative Algebra

Spaces and Functions on Spaces

Associated Primes

Dimension, Depth and Cohen-Macaulay Modules

Krull Dimension

Slicing Dimension

Hilbert-Samuel Dimension

Weierstrass Dimension

The Hauptidealsatz

Depth and Cohen-Macaulay Modules

Free Resolutions

Cohen-Macaulay Modules and Freeness

Examples of Cohen-Macaulay Spaces

Pulling Back Algebraic Structures

Samuel Multiplicity

Local Analytic Geometrylay? The Preparation Theorem

Local Properties of Analytic Sets and Finite Mappings

Degree and Multiplicity

Normalisation of Analytic Set-Germs

Extension Theorems

Normalisation

Sheaves

Presheaves and Sheaves

Coherence

Conservation of Multiplicity

Representatives

Conservation of Multiplicity II

References

Index.

Notes:

Includes bibliographical references (pages 553-562) and index.

Current copyright fee: GBP19.00 42\0.

ISBN:

3030344398

9783030344399

OCLC:

1140511543