The Disc Embedding TheoremStefan Behrens, Boldizsár Kalmár, Min Hoon Kim, Mark Powell, Arunima Ray 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
Context for the Disc Embedding Theorem | 1 |
Outline of the Upcoming Proof | 27 |
Decomposition Space Theory | 43 |
Building Skyscrapers | 153 |
Interlude | 271 |
Skyscrapers Are Standard | 383 |
| 457 | |
| 469 | |
Other editions - View all
The Disc Embedding Theorem Stefan Behrens,Boldizsar Kalmar,Min Hoon Kim,Mark Powell,Arunima Ray Limited preview - 2021 |
The Disc Embedding Theorem Stefan Behrens,Boldizsar Kalmar,Min Hoon Kim,Mark Powell,Arunima Ray Limited preview - 2021 |
Common terms and phrases
1-storey capped tower 2-handle 4-dimensional ambient approximable by homeomorphisms Arunima Ray attaching circle attaching region ball capped grope capped surfaces Casson handles Chapter closed collar compact complement conjecture connected components construction corresponding decomposition elements decomposition space definition degree one normal denote diffeomorphic disc embedding theorem double point loops dual spheres embedded discs finite binary word fundamental group geometrically dual geometrically transverse spheres h-cobordism theorem handle decomposition homeomorphism homotopy equivalence immersed discs infinite intersection form intersection points isotopy Kirby diagram Kirby-Siebenmann invariant manifold Mark Powell meridian metric space mutually disjoint neighbourhood normal bundle normal map null pair pairwise disjoint parallel copies Poincaré conjecture Proposition quotient map ramified Bing-Whitehead result Section shrink shrinkable simply connected Sing(r skyscraper slice solid tori starlike-equivalent storey subset surface stages tip region topological 4-manifold torus trivial Whitehead doubling Whitney discs Whitney move π₁