 Title Pages
 Preface
 List of figures

1 Context for the Disc Embedding Theorem 
2 Outline of the Upcoming Proof 
Part I Decomposition Space Theory 
3 The Schoenflies Theorem after Mazur, Morse, and Brown 
4 Decomposition Space Theory and the Bing Shrinking Criterion 
5 The Alexander Gored Ball and the Bing Decomposition 
6 A Decomposition That Does Not Shrink 
7 The Whitehead Decomposition 
8 Mixed Bing–Whitehead Decompositions 
9 Shrinking Starlike Sets 
10 The Ball to Ball Theorem 
Part II Building Skyscrapers 
11 Intersection Numbers and the Statement of the Disc Embedding Theorem 
12 Gropes, Towers, and Skyscrapers 
13 Picture Camp 
14 Architecture of Infinite Towers and Skyscrapers 
15 Basic Geometric Constructions 
16 From Immersed Discs to Capped Gropes 
17 Grope Height Raising and 1storey Capped Towers 
18 Tower Height Raising and Embedding 
Part III Interlude 
19 Good Groups 
20 The scobordism Theorem, the Sphere Embedding Theorem, and the Poincaré Conjecture 
21 The Development of Topological 4manifold Theory 
22 Surgery Theory and the Classification of Closed, Simply Connected 4manifolds 
23 Open Problems 
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 
27 Skyscrapers Are Standard: An Overview 
28 Skyscrapers Are Standard: The Details  Afterword: PC4 at Age 40
 References
 Index
The Ball to Ball Theorem
The Ball to Ball Theorem
 Chapter:
 (p.131) 10 The Ball to Ball Theorem
 Source:
 The Disc Embedding Theorem
 Author(s):
Stefan Behrens
Boldizsár Kalmár
Daniele Zuddas
 Publisher:
 Oxford University Press
The ball to ball theorem is presented, which states that a map from the 4ball to itself, restricting to a homeomorphism on the 3sphere, whose inverse sets are null and have nowhere dense image, is approximable by homeomorphisms relative to the boundary. The approximating homeomorphisms are produced abstractly, as in the previous chapter, with no need to investigate the decomposition elements further. In the proof of the disc embedding theorem, a decomposition of the 4ball will be constructed, called the gaps^{+} decomposition. The ball to ball theorem will be used to prove that this decomposition shrinks; this is called the βshrink.
Keywords: ball to ball theorem, null decomposition, βshrink, disc embedding theorem, gaps+ decomposition
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 Title Pages
 Preface
 List of figures

1 Context for the Disc Embedding Theorem 
2 Outline of the Upcoming Proof 
Part I Decomposition Space Theory 
3 The Schoenflies Theorem after Mazur, Morse, and Brown 
4 Decomposition Space Theory and the Bing Shrinking Criterion 
5 The Alexander Gored Ball and the Bing Decomposition 
6 A Decomposition That Does Not Shrink 
7 The Whitehead Decomposition 
8 Mixed Bing–Whitehead Decompositions 
9 Shrinking Starlike Sets 
10 The Ball to Ball Theorem 
Part II Building Skyscrapers 
11 Intersection Numbers and the Statement of the Disc Embedding Theorem 
12 Gropes, Towers, and Skyscrapers 
13 Picture Camp 
14 Architecture of Infinite Towers and Skyscrapers 
15 Basic Geometric Constructions 
16 From Immersed Discs to Capped Gropes 
17 Grope Height Raising and 1storey Capped Towers 
18 Tower Height Raising and Embedding 
Part III Interlude 
19 Good Groups 
20 The scobordism Theorem, the Sphere Embedding Theorem, and the Poincaré Conjecture 
21 The Development of Topological 4manifold Theory 
22 Surgery Theory and the Classification of Closed, Simply Connected 4manifolds 
23 Open Problems 
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 
27 Skyscrapers Are Standard: An Overview 
28 Skyscrapers Are Standard: The Details  Afterword: PC4 at Age 40
 References
 Index