The Disc Embedding Theorem

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.

• Preface1: Context for the disc embedding theorem2: Outline of the upcoming proofPart 1: Decomposition space theory3: The Schoenflies theorem after Mazur, Morse, and Brown4: Decomposition space theory and the Bing shrinking criterion5: The Alexander gored ball and the Bing decomposition6: A decomposition that does not shrink7: The Whitehead decomposition8: Mixed Bing-Whitehead decompositions9: Shrinking starlike sets10: The ball to ball theoremPart II: Building skyscrapers11: Intersection numbers and the statement of the disc embedding theorem12: Gropes, towers, and skyscrapers13: Picture camp14: Architecture of infinite towers and skyscrapers15: Basic geometric constructions16: From immersed discs to capped gropes17: Grope height raising and 1-storey capped towers18: Tower height raising and embeddingPart III: Interlude19: Good groups20: The s-cobordism theorem, the sphere embedding theorem, and the Poincaré conjecture21: The development of topological 4-manifold theory22: Surgery theory and the classification of closed, simply connected 4-manifolds23: Open problemsPart IV: Skyscrapers are standard24: Replicable rooms and boundary shrinkable skyscrapers25: The collar adding lemma26: Key facts about skyscrapers and decomposition space theory27: Skyscrapers are standard: an overview28: Skyscrapers are standard: the detailsBibliographyAfterwordIndex

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