The Disc Embedding Theorem
Based on the work of Michael H. Freedman
Published on 20. July 2021
Book
Hardback
496 pages
978-0-19-884131-9 (ISBN)
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 Poincare 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.
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.
Reviews / Votes
The authors take enormous pains to make the arguments as detailed and self-contained as possible, including hundreds of beautifully rendered diagrams that render complicated geometric arguments sensible. This book has arrived at just the right time * Danny C. Calegari, MathSciNet *More details
Language
English
Place of publication
Oxford
United Kingdom
Target group
Professional and scholarly
Illustrations
220
Dimensions
Height: 243 mm
Width: 160 mm
Thickness: 30 mm
Weight
944 gr
ISBN-13
978-0-19-884131-9 (9780198841319)
Copyright in bibliographic data and cover images is held by Nielsen Book Services Limited or by the publishers or by their respective licensors: all rights reserved.
Schweitzer Classification
Other editions
Additional editions
Stefan Behrens | Boldizsar Kalmar | Min Hoon Kim
The Disc Embedding Theorem
E-Book
07/2021
1st Edition
OUP eBook
€98.99
Available for download
Persons
Dr Stefan Behrens is an assistant professor in the Geometry and Topology group led by Prof. Dr. Stefan Bauer. His field of research is low dimensional topology, with a focus on the topology of smooth 4-manifolds.
Boldizsar Kalmar was a research assistant at the Alfred Renyi Institute of Mathematics in 2005, then he got his PhD at Kyushu University in Japan in 2008. Then he did research at the Alfred Renyi Institute of Mathematics until 2017. He visited the Max Planck Institute for Mathematics in 2013. His research field is the topology of singular maps and low dimensional topology.
Dr Mark Powell obtained his PhD from the University of Edinburgh under the supervision of Andrew Ranicki in 2011. After positions at Indiana University, the Max Planck Institute in Bonn, and at UQAM in Montreal, he moved to Durham University in 2017 to take up a position as Associate Professor.
Dr Arunima Ray received a PhD in mathematics from Rice University, in Houston, USA in 2014 and subsequently held a postdoctoral fellowship at Brandeis University at Waltham, USA. She is currently a Lise Meitner group leader at the Max Planck Institute for Mathematics in Bonn, Germany. Her research is in low-dimensional topology, specifically the study of knots and links, and 3- and 4-manifolds.
Boldizsar Kalmar was a research assistant at the Alfred Renyi Institute of Mathematics in 2005, then he got his PhD at Kyushu University in Japan in 2008. Then he did research at the Alfred Renyi Institute of Mathematics until 2017. He visited the Max Planck Institute for Mathematics in 2013. His research field is the topology of singular maps and low dimensional topology.
Dr Mark Powell obtained his PhD from the University of Edinburgh under the supervision of Andrew Ranicki in 2011. After positions at Indiana University, the Max Planck Institute in Bonn, and at UQAM in Montreal, he moved to Durham University in 2017 to take up a position as Associate Professor.
Dr Arunima Ray received a PhD in mathematics from Rice University, in Houston, USA in 2014 and subsequently held a postdoctoral fellowship at Brandeis University at Waltham, USA. She is currently a Lise Meitner group leader at the Max Planck Institute for Mathematics in Bonn, Germany. Her research is in low-dimensional topology, specifically the study of knots and links, and 3- and 4-manifolds.
Editor
Assistant ProfessorAssistant Professor, Bielefeld University
Assistant ProfessorAssistant Professor, Budapest University of Technology and Economics
Assistant ProfessorAssistant Professor, Chonnam National University
Durham UniversityDurham University, Associate Professor
Lise Meitner group leaderLise Meitner group leader, Max Planck Institute for Mathematics
Content
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 Poincare 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
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 Poincare 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