The disc embedding theorem / editors, Stefan Behrens [et al.]

Format:

Book

Author/Creator:

Behrens, Stefan, author.

Series:

Oxford scholarship online.

Oxford scholarship online

Language:

English

Subjects (All):

Differential topology.

Mathematics--Study and teaching.

Mathematics.

Genre:

Instructional and educational works.

Physical Description:

1 online resource (xvii, 473 pages) : illustrations (black and white, and colour).

Edition:

First edition.

Place of Publication:

Oxford, UK : Oxford University Press, [2021]

Summary:

This text contains the first thorough and approachable exposition of Freedman's proof of the disc embedding theorem.

Contents:

Cover

The Disc Embedding Theorem

Copyright

Preface

The Origin of This Book

Casson Towers

Differences

Seminar Organization

Credit

Contents

List of Figures

1: Context for the Disc Embedding Theorem

1.1 Before the Disc Embedding Theorem

1.1.1 High-dimensional Surgery Theory

1.1.2 Attempting 4-dimensional Surgery

1.1.3 Attempting to Prove the s-cobordism Theorem

1.2 The Whitney Move in Dimension Four

1.3 Casson's Insight: Geometric Duals

1.3.1 Surgery and Geometric Duals

1.3.2 The s-cobordism Theorem and Geometric Duals

1.4 Casson Handles

1.5 The Disc Embedding Theorem

1.6 After the Disc Embedding Theorem

1.6.1 Foundational Results

1.6.2 Classification Results

1.6.3 Knot Theory Results

2: Outline of the Upcoming Proof

2.1 Preparation

2.2 Building Skyscrapers

2.3 Skyscrapers Are Standard

2.4 Reader's Guide

Part I: Decomposition Space Theory

3: The Schoenflies Theorem after Mazur, Morse, and Brown

3.1 Mazur's Theorem

3.2 Morse's Theorem

3.3 Shrinking Cellular Sets

3.4 Brown's Proof of the Schoenflies Theorem

4: Decomposition Space Theory and the Bing Shrinking Criterion

4.1 The Bing Shrinking Criterion

4.2 Decompositions

4.3 Upper Semi-continuous Decompositions

4.4 Shrinkability of Decompositions

5: The Alexander Gored Ball and the Bing Decomposition

5.1 Three Descriptions of the Alexander Gored Ball

5.1.1 An Intersection of 3-balls in D3

5.1.2 A (3-dimensional) Grope

5.1.3 A Decomposition Space

5.2 The Bing Decomposition: The First Ever Shrink

6: A Decomposition That Does Not Shrink

7: The Whitehead Decomposition

7.1 The Whitehead Decomposition Does Not Shrink

7.2 The Space S3/W Is a Manifold Factor

8: Mixed Bing-Whitehead Decompositions

8.1 Toroidal Decompositions.

8.2 Disc Replicating Functions

8.3 Shrinking of Toroidal Decompositions

8.4 Computing the Disc Replicating Function

9: Shrinking Starlike Sets

9.1 Null Collections and Starlike Sets

9.2 Shrinking Null, Recursively Starlike-equivalent Decompositions

9.3 Literature Review

10: The Ball to Ball Theorem

10.1 The Main Idea of the Proof

10.2 Relations

10.3 Admissible Diagrams and the Main Lemma

10.4 Proof of the Ball to Ball Theorem

10.5 The General Position Lemma

10.6 The Sphere to Sphere Theorem from the Ball to Ball Theorem

Part II: Building Skyscrapers

11: Intersection Numbers and the Statement of the Disc Embedding Theorem

11.1 Immersions

11.2 Whitney Moves and Finger Moves

11.2.1 Whitney Moves

11.2.2 Finger Moves

11.3 Intersection and Self-intersection Numbers

11.4 Statement of the Disc Embedding Theorem

12: Gropes, Towers, and Skyscrapers

12.1 Gropes and Towers

12.2 Infinite Towers and Skyscrapers

13: Picture Camp

13.1 Dehn Surgery

13.2 Kirby Diagrams

13.2.1 Attaching 1-handles

13.2.2 Attaching 2-handles

13.2.3 Attaching and Tip Regions

13.3 Kirby Calculus

13.3.1 Handle Slides

13.3.2 Handle Cancellation

13.3.3 Plumbing

13.4 Kirby Diagrams for Generalized Towers

13.4.1 Surface Blocks

13.4.2 Disc and Cap Blocks

13.4.3 Stages

13.4.4 Generalized Towers

13.5 Bing and Whitehead Doubling

13.6 Simplification

13.6.1 Bing Doubles

13.6.2 Whitehead Doubles

13.6.3 Trees Associated with Generalized Towers

13.6.4 Kirby Diagrams from Trees

13.7 Chapter Summary

14: Architecture of Infinite Towers and Skyscrapers

14.1 Infinite Towers

14.2 Infinite Compactified Towers

14.3 Skyscrapers

15: Basic Geometric Constructions

15.1 The Clifford Torus

15.2 Elementary Geometric Techniques

15.2.1 Tubing.

15.2.2 Boundary Twisting

15.2.3 Making Whitney Circles Disjoint

15.2.4 Pushing Down Intersections

15.2.5 Contraction and Subsequent Pushing Off

15.3 Replacing Algebraic Duals with Geometric Duals

16: From Immersed Discs to Capped Gropes

17: Grope Height Raising and 1-storey Capped Towers

17.1 Grope Height Raising

17.2 1-storey Capped Towers

17.3 Continuation of the Proof of the Disc Embedding Theorem

18: Tower Height Raising and Embedding

18.1 The Tower Building Permit

18.2 The Tower Squeezing Lemma

18.3 The Tower and Skyscraper Embedding Theorems

18.4 Proof of the Disc Embedding Theorem, Assuming Part IV

Part III: Interlude

19: Good Groups

20: The s-cobordism Theorem, the Sphere Embedding Theorem, and the Poincaré Conjecture

20.1 The s-cobordism Theorem

20.2 The Poincaré Conjecture

20.3 The Sphere Embedding Theorem

21: The Development of Topological 4-manifold Theory

21.1 Results Proven in This Book

21.2 Input to the Flowchart

21.2.1 Immersion Theory and Smoothing Noncompact, Contractible 4-manifolds

21.2.2 Donaldson Theory

21.3 Further Results from Freedman's Original Paper

21.3.1 The Proper h-cobordism Theorem with Smooth Input

21.3.2 Integral Homology 3-spheres Bound Contractible 4-manifolds

21.4 Foundational Results Due to Quinn

21.4.1 The Controlled h-cobordism Theorem with Smooth Input

21.4.2 Handle Straightening

21.4.3 The Stable Homeomorphism Theorem

21.4.4 The Annulus Theorem and Connected Sum

21.4.5 The Sum Stable Smoothing Theorem

21.4.6 TOP(4)=O(4)!TOP=O is 5-connected

21.4.7 Smoothing Away from a Point

21.4.8 Normal Bundles

21.4.9 Topological Transversality and Map Transversality

21.4.10 The Immersion Lemma

21.4.11 Handle Decompositions of 5-manifolds

21.5 Category Preserving Theorems.

21.5.1 The Surgery Sequence for Good Groups

21.6 Flagship Results

21.6.1 Classification of Closed, Simply Connected 4-manifolds

21.6.2 The Poincaré Conjecture with Topological Input

21.6.3 Alexander Polynomial One Knots Are Slice

21.6.4 Slice Knots

21.6.5 Exotic R4s

21.6.6 Computation of the 4-dimensional Bordism Group

22: Surgery Theory and the Classification of Closed, Simply Connected 4-manifolds

22.1 The Surgery Sequence

22.1.1 Normal Maps

22.1.2 L-groups

22.1.3 The Surgery Obstruction Map

22.1.4 Exactness at the Normal Maps

22.1.5 Wall Realization

22.1.6 Exactness at the Structure Set

22.2 The Surgery Sequence for Manifolds with Boundary

22.3 Classification of Closed, Simply Connected 4-manifolds

22.3.1 Existence of a 4-manifold

22.3.2 Size of the Structure Set

22.3.3 Realizing Isometries by Homeomorphisms

22.3.4 Other Homeomorphism Classifications of 4-manifolds

23: Open Problems

23.1 The Disc Embedding Conjecture

23.1.1 Standard Slices for Universal Links

23.2 The Surgery Conjecture

23.2.1 Good Boundary Links

23.2.2 Free Slice Discs and the Link Family L1

23.2.3 Free Slice Discs and the Link Family L2

23.2.4 The A-B Slice Problem

23.3 The s-cobordism Conjecture

23.3.1 Round Handles

Part IV: Skyscrapers Are Standard

24: Replicable Rooms and Boundary Shrinkable Skyscrapers

25: The Collar Adding Lemma

26: Key Facts about Skyscrapers and Decomposition Space Theory

26.1 Ingredients from Part I

26.2 Ingredients from Part II

27: Skyscrapers Are Standard: An Overview

27.1 An Outline of the Strategy

27.2 The Strategy in More Detail

27.3 Some Things to Keep in Mind

28: Skyscrapers Are Standard: The Details

28.1 Binary Words and the Cantor Set

28.2 The Standard Handle

28.3 Embedding the Design in a Skyscraper.

28.3.1 The Design Piece for the Empty Word

28.3.2 Design Pieces for Finite Binary Words

28.3.3 Design Pieces for Infinite Binary Words

28.4 Embedding the Design in the Standard Handle

28.4.1 Embedding Finite Word Design Pieces in H

28.4.2 Embedding Infinite Word Design Pieces in H

28.5 From Holes and Gaps to Holes+ and Gaps+

28.6 Shrinking the Complement of the Design

28.6.1 The Common Quotient

28.6.2 The Map Is Approximable by Homeomorphisms

28.6.3 The Map Is Approximable by Homeomorphisms

Afterword: PC4 at Age 40

References

Index.

Notes:

This edition also issued in print: 2021.

Includes bibliographical references and index.

Description based on print version record.

Local Notes:

Case binding.

ISBN:

0-19-187692-5

0-19-257838-3

OCLC:

1259590730