The Disc Embedding Theorem

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.

Table of 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