| Description |
1 online resource (xvii, 473 pages) : illustrations (some color) |
| Contents |
Cover -- The Disc Embedding Theorem -- Copyright -- Preface -- The Origin of This Book -- Casson Towers -- Differences -- Seminar Organization -- Credit -- Contents -- List of Figures -- 1: Context for the Disc Embedding Theorem -- 1.1 Before the Disc Embedding Theorem -- 1.1.1 High-dimensional Surgery Theory -- 1.1.2 Attempting 4-dimensional Surgery -- 1.1.3 Attempting to Prove the s-cobordism Theorem -- 1.2 The Whitney Move in Dimension Four -- 1.3 Casson's Insight: Geometric Duals -- 1.3.1 Surgery and Geometric Duals -- 1.3.2 The s-cobordism Theorem and Geometric Duals |
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1.4 Casson Handles -- 1.5 The Disc Embedding Theorem -- 1.6 After the Disc Embedding Theorem -- 1.6.1 Foundational Results -- 1.6.2 Classification Results -- 1.6.3 Knot Theory Results -- 2: Outline of the Upcoming Proof -- 2.1 Preparation -- 2.2 Building Skyscrapers -- 2.3 Skyscrapers Are Standard -- 2.4 Reader's Guide -- Part I: Decomposition Space Theory -- 3: The Schoenflies Theorem after Mazur, Morse, and Brown -- 3.1 Mazur's Theorem -- 3.2 Morse's Theorem -- 3.3 Shrinking Cellular Sets -- 3.4 Brown's Proof of the Schoenflies Theorem |
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4: Decomposition Space Theory and the Bing Shrinking Criterion -- 4.1 The Bing Shrinking Criterion -- 4.2 Decompositions -- 4.3 Upper Semi-continuous Decompositions -- 4.4 Shrinkability of Decompositions -- 5: The Alexander Gored Ball and the Bing Decomposition -- 5.1 Three Descriptions of the Alexander Gored Ball -- 5.1.1 An Intersection of 3-balls in D3 -- 5.1.2 A (3-dimensional) Grope -- 5.1.3 A Decomposition Space -- 5.2 The Bing Decomposition: The First Ever Shrink -- 6: A Decomposition That Does Not Shrink -- 7: The Whitehead Decomposition -- 7.1 The Whitehead Decomposition Does Not Shrink |
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7.2 The Space S3/W Is a Manifold Factor -- 8: Mixed Bing-Whitehead Decompositions -- 8.1 Toroidal Decompositions -- 8.2 Disc Replicating Functions -- 8.3 Shrinking of Toroidal Decompositions -- 8.4 Computing the Disc Replicating Function -- 9: Shrinking Starlike Sets -- 9.1 Null Collections and Starlike Sets -- 9.2 Shrinking Null, Recursively Starlike-equivalent Decompositions -- 9.3 Literature Review -- 10: The Ball to Ball Theorem -- 10.1 The Main Idea of the Proof -- 10.2 Relations -- 10.3 Admissible Diagrams and the Main Lemma -- 10.4 Proof of the Ball to Ball Theorem |
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10.5 The General Position Lemma -- 10.6 The Sphere to Sphere Theorem from the Ball to Ball Theorem -- Part II: Building Skyscrapers -- 11: Intersection Numbers and the Statement of the Disc Embedding Theorem -- 11.1 Immersions -- 11.2 Whitney Moves and Finger Moves -- 11.2.1 Whitney Moves -- 11.2.2 Finger Moves -- 11.3 Intersection and Self-intersection Numbers -- 11.4 Statement of the Disc Embedding Theorem -- 12: Gropes, Towers, and Skyscrapers -- 12.1 Gropes and Towers -- 12.2 Infinite Towers and Skyscrapers -- 13: Picture Camp -- 13.1 Dehn Surgery -- 13.2 Kirby Diagrams |
| Summary |
The Disc Embedding Theorem contains the first thorough and approachable exposition of Freedman's proof of the disc embedding theorem |
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The disc embedding theorem provides a detailed proof of the eponymous theorem in 4-manifold topology. The theorem, due to Michael Freedman, underpins virtually all of our understanding of 4-manifolds in the topological category. Most famously, this includes the 4-dimensional topological Poincaré conjecture. Combined with the concurrent work of Simon Donaldson, the theorem reveals a remarkable disparity between the topological and smooth categories for 4-manifolds. A thorough exposition of Freedman's proof of the disc embedding theorem is given, 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. Techniques from decomposition space theory are used to show that an object produced by an infinite, iterative process, which we call a skyscraper, is homeomorphic to a thickened disc, relative to its boundary. A stand-alone interlude explains the disc embedding theorem's key role in smoothing theory, the existence of exotic smooth structures on Euclidean space, and all known homeomorphism classifications of 4-manifolds via surgery theory and the s-cobordism theorem. The book is written to be accessible to graduate students working on 4-manifolds, as well as researchers in related areas. It contains over a hundred professionally rendered figures |
| Bibliography |
Includes bibliographical references and index |
| Notes |
Description based on online resource; title from PDF title page (Oxford Scholarship Online, viewed April 21, 2022) |
| Subject |
Differential topology.
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Differential topology
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| Form |
Electronic book
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| Author |
Behrens, Stefan (Mathematician), editor.
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Kalmár, Boldizsár, editor
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Kim, Min Hoon, editor
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Powell, Mark, (Mathematician), editor.
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Ray, Arunima, editor
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| ISBN |
9780192578389 |
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0192578383 |
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9780191876929 |
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0191876925 |
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