The disc embedding theorem / edited by Stefan Behrens, Boldizsár Kalmár, Min Hoon Kim, Mark Powell, and Arunima Ray.

Format:

Book

Contributor:

Behrens, Stefan (Mathematician), editor.

Kalmár, Boldizsár, editor.

Kim, Min Hoon, editor.

Powell, Mark, (Mathematician), editor.

Ray, Arunima, editor.

Language:

English

Subjects (All):

Differential topology.

Embedding theorems.

Four-manifolds (Topology).

Physical Description:

xvii, 473 pages : illustrations ; 24 cm

Place of Publication:

Oxford : Oxford University Press, 2021.

Summary:

"Based on the 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 book contains the first thorough and accessible 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 nondifferentiable homeomorphisms between manifolds, is provided, as well as a stand-along 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 theorim. 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."--Back cover

Contents:

Machine generated contents note: 1. Context for the Disc Embedding Theorem / Arunima Ray

2. Outline of the Upcoming Proof / Arunima Ray

pt. I DECOMPOSITION SPACE THEORY

3. The Schoenflies Theorem after Mazur, Morse, and Brown / Peter Teichner

4. Decomposition Space Theory and the Bing Shrinking Criterion / Henrik Ruping

5. The Alexander Gored Ball and the Bing Decomposition / Min Hoon Kim

6. A Decomposition That Does Not Shrink / Mark Powell

7. The Whitehead Decomposition / Nathan Sunukjian

8. Mixed Bing-Whitehead Decompositions / Min Hoon Kim

9. Shrinking Starlike Sets / Arunima Ray

10. The Ball to Ball Theorem / Daniele Zuddas

pt. II BUILDING SKYSCRAPERS

11. Intersection Numbers and the Statement of the Disc Embedding Theorem / Arunima Ray

12. Gropes, Towers, and Skyscrapers / Arunima Ray

13. Picture Camp / Arunima Ray

14. Architecture of Infinite Towers and Skyscrapers / Mark Powell

15. Basic Geometric Constructions / Arunima Ray

16. From Immersed Discs to Capped Gropes / Arunima Ray

17. Grope Height Raising and 1-storey Capped Towers / Mark Powell

18. Tower Height Raising and Embedding / Mark Powell

pt. III INTERLUDE

19. Good Groups / Arunima Ray

20. The s-cobordism Theorem, the Sphere Embedding Theorem, and the Poincare Conjecture / Arunima Ray

21. The Development of Topological 4-manifold Theory / Arunima Ray

22. Surgery Theory and the Classification of Closed, Simply Connected 4-manifolds / Arunima Ray

23. Open Problems / Arunima Ray

pt. IV SKYSCRAPERS ARE STANDARD

24. Replicable Rooms and Boundary Shrinkable Skyscrapers / Mark Powell

25. The Collar Adding Lemma / Mark Powell

26. Key Facts about Skyscrapers and Decomposition Space Theory / Arunima Ray

27. Skyscrapers Are Standard: An Overview / Stefan Behrens

28. Skyscrapers Are Standard: The Details / Arunima Ray.

Notes:

Includes bibliographical references and index.

ISBN:

9780198841319

0198841310

OCLC:

1241731061

Publisher Number:

99988778375