Based on Fields medal winning work of Michael Freedman, this book explores the disc embedding theorem for 4-dimensional manifolds. This theorem underpins virtually all our understanding of topological 4-manifolds. Most famously, this includes the 4-dimensional Poincaré conjecture in the topological category.
The Disc Embedding Theorem contains the first thorough and approachable exposition of Freedman's proof of the disc embedding theorem, with many new details. A self-contained account of decomposition space theory, a beautiful but outmoded branch of topology that produces non-differentiable homeomorphisms between manifolds, is provided, as well as a stand-alone interlude that explains the disc embedding theorem's key role in all known homeomorphism classifications of 4-manifolds via
surgery theory and the s-cobordism theorem. Additionally, the ramifications of the disc embedding theorem within the study of topological 4-manifolds, for example Frank Quinn's development of fundamental tools like transversality are broadly described.
The book is written for mathematicians, within the subfield of topology, specifically interested in the study of 4-dimensional spaces, and includes numerous professionally rendered figures.
Author(s): Stefan Behrens, Boldizsar Kalmar, Min Hoon Kim, Mark Powell, Arunima Ray
Publisher: Oxford University Press
Year: 2021
Language: English
Pages: 473
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
