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.

• 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 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 Bibliography Afterword Index

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