ISBN : 9780198841319
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 Poincare 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.
Preface
1 Context for the disc embedding theorem
2 Outline of the upcoming proof
Part 1: Decomposition space theory
3 The Schoenies 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 1-storey capped towers
18 Tower height raising and embedding
Part III: Interlude
19 Good groups
20 The s-cobordism theorem, the sphere embedding theorem, and the Poincare conjecture
21 The development of topological 4-manifold theory
22 Surgery theory and the classification of closed, simply connected 4-manifolds
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
Bibliography
Afterword
Index
