The Disc Embedding Theorem Stefan Behrens 0198841310 9780198841319

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Product details:

• ISBN-10: 0198841310
• ISBN-13‏: ‎9780198841319
• Author: Stefan Behrens

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.

Table of contents:

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

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