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Full Description
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.
Contents
Preface
1: Context for the disc embedding theorem
2: Outline of the upcoming proof
Part 1: 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 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 Poincaré 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
