Pub Date : 2021-07-20DOI: 10.1093/oso/9780198841319.003.0007
Xiaoyi Cui, B. Kalmár, P. Orson, Nathan Sunukjian
‘The Whitehead Decomposition’ introduces this historically significant decomposition. Not only is the quotient of the 3-sphere by the Whitehead decomposition not homeomorphic to the 3-sphere, it is not even a manifold. In order to detect this curious fact, the notion of a noncompact space being simply connected at infinity is introduced. The chapter also describes the Whitehead manifold, which is a contractible 3-manifold not homeomorphic to Euclidean space. While the Whitehead decomposition does not shrink, its product with the real line does, as is proved in this chapter; in other words, the quotient of the 3-sphere by the Whitehead decomposition is a manifold factor. The proof of the disc embedding theorem utilizes Bing–Whitehead decompositions, which may be understood to be a mix between the Whitehead decomposition and the Bing decomposition from a previous chapter. In a subsequent chapter, precisely when Bing–Whitehead decompositions shrink is explained.
{"title":"The Whitehead Decomposition","authors":"Xiaoyi Cui, B. Kalmár, P. Orson, Nathan Sunukjian","doi":"10.1093/oso/9780198841319.003.0007","DOIUrl":"https://doi.org/10.1093/oso/9780198841319.003.0007","url":null,"abstract":"‘The Whitehead Decomposition’ introduces this historically significant decomposition. Not only is the quotient of the 3-sphere by the Whitehead decomposition not homeomorphic to the 3-sphere, it is not even a manifold. In order to detect this curious fact, the notion of a noncompact space being simply connected at infinity is introduced. The chapter also describes the Whitehead manifold, which is a contractible 3-manifold not homeomorphic to Euclidean space. While the Whitehead decomposition does not shrink, its product with the real line does, as is proved in this chapter; in other words, the quotient of the 3-sphere by the Whitehead decomposition is a manifold factor. The proof of the disc embedding theorem utilizes Bing–Whitehead decompositions, which may be understood to be a mix between the Whitehead decomposition and the Bing decomposition from a previous chapter. In a subsequent chapter, precisely when Bing–Whitehead decompositions shrink is explained.","PeriodicalId":272723,"journal":{"name":"The Disc Embedding Theorem","volume":"192 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127358268","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-20DOI: 10.1093/oso/9780198841319.003.0010
Stefan Behrens, B. Kalmár, D. Zuddas
The ball to ball theorem is presented, which states that a map from the 4-ball to itself, restricting to a homeomorphism on the 3-sphere, whose inverse sets are null and have nowhere dense image, is approximable by homeomorphisms relative to the boundary. The approximating homeomorphisms are produced abstractly, as in the previous chapter, with no need to investigate the decomposition elements further. In the proof of the disc embedding theorem, a decomposition of the 4-ball will be constructed, called the gaps+ decomposition. The ball to ball theorem will be used to prove that this decomposition shrinks; this is called the β-shrink.
{"title":"The Ball to Ball Theorem","authors":"Stefan Behrens, B. Kalmár, D. Zuddas","doi":"10.1093/oso/9780198841319.003.0010","DOIUrl":"https://doi.org/10.1093/oso/9780198841319.003.0010","url":null,"abstract":"The ball to ball theorem is presented, which states that a map from the 4-ball to itself, restricting to a homeomorphism on the 3-sphere, whose inverse sets are null and have nowhere dense image, is approximable by homeomorphisms relative to the boundary. The approximating homeomorphisms are produced abstractly, as in the previous chapter, with no need to investigate the decomposition elements further. In the proof of the disc embedding theorem, a decomposition of the 4-ball will be constructed, called the gaps+ decomposition. The ball to ball theorem will be used to prove that this decomposition shrinks; this is called the β-shrink.","PeriodicalId":272723,"journal":{"name":"The Disc Embedding Theorem","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125961266","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-20DOI: 10.1093/oso/9780198841319.003.0028
Stefan Behrens, Daniel Kasprowski, Mark D. Powell, Arunima Ray
‘Skyscrapers Are Standard: The Details’ provides a thorough and detailed proof that every skyscraper is homeomorphic to the standard 2-handle, relative to the attaching region. Results from decomposition space theory established in Part I and the constructive results from Part II are combined. The idea is to construct a subset of a skyscraper called the design, define an embedding of this subset into the standard 2-handle, and then consider the decomposition spaces obtained by quotienting out the connected components of the complement of this common subset. It is shown that the decomposition spaces are homeomorphic, and that both quotient maps are approximable by homeomorphisms. This chapter also shows that everything can be done fixing a neighbourhood of the attaching region. It is then deduced that skyscrapers are standard, as desired.
{"title":"Skyscrapers Are Standard: The Details","authors":"Stefan Behrens, Daniel Kasprowski, Mark D. Powell, Arunima Ray","doi":"10.1093/oso/9780198841319.003.0028","DOIUrl":"https://doi.org/10.1093/oso/9780198841319.003.0028","url":null,"abstract":"‘Skyscrapers Are Standard: The Details’ provides a thorough and detailed proof that every skyscraper is homeomorphic to the standard 2-handle, relative to the attaching region. Results from decomposition space theory established in Part I and the constructive results from Part II are combined. The idea is to construct a subset of a skyscraper called the design, define an embedding of this subset into the standard 2-handle, and then consider the decomposition spaces obtained by quotienting out the connected components of the complement of this common subset. It is shown that the decomposition spaces are homeomorphic, and that both quotient maps are approximable by homeomorphisms. This chapter also shows that everything can be done fixing a neighbourhood of the attaching region. It is then deduced that skyscrapers are standard, as desired.","PeriodicalId":272723,"journal":{"name":"The Disc Embedding Theorem","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121667627","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-20DOI: 10.1093/oso/9780198841319.003.0015
Mark Powell, Arunima Ray
Basic geometric constructions, including tubing, boundary twisting, pushing down intersections, and contraction followed by push-off are presented. These moves are used repeatedly later in the proof. New, detailed pictures illustrating these constructions are provided. The Clifford torus at an intersection point between two surfaces in 4-dimensional space is described. The chapter closes with an important application of some of these moves called the Geometric Casson Lemma. This lemma upgrades algebraically dual spheres to geometrically dual spheres, at the cost of introducing more self-intersections. It is also shown that an immersed Whitney move is a regular homotopy of the associated surfaces.
{"title":"Basic Geometric Constructions","authors":"Mark Powell, Arunima Ray","doi":"10.1093/oso/9780198841319.003.0015","DOIUrl":"https://doi.org/10.1093/oso/9780198841319.003.0015","url":null,"abstract":"Basic geometric constructions, including tubing, boundary twisting, pushing down intersections, and contraction followed by push-off are presented. These moves are used repeatedly later in the proof. New, detailed pictures illustrating these constructions are provided. The Clifford torus at an intersection point between two surfaces in 4-dimensional space is described. The chapter closes with an important application of some of these moves called the Geometric Casson Lemma. This lemma upgrades algebraically dual spheres to geometrically dual spheres, at the cost of introducing more self-intersections. It is also shown that an immersed Whitney move is a regular homotopy of the associated surfaces.","PeriodicalId":272723,"journal":{"name":"The Disc Embedding Theorem","volume":"88 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131773413","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-20DOI: 10.1093/oso/9780198841319.003.0026
Mark Powell, Arunima Ray
‘Key Facts about Skyscrapers and Decomposition Space Theory’ summarizes the input from Parts I and II needed for the remainder of the proof of the disc embedding theorem. Precise references to previous chapters are provided. This enables Part IV to be read independently from the previous parts, provided the reader is willing to accept the facts from Parts I and II summarized in this chapter. The listed facts include the shrinking of mixed, ramified Bing–Whitehead decompositions of the solid torus, the shrinking of null decompositions consisting of recursively starlike-equivalent sets, the ball to ball theorem, the skyscraper embedding theorem, and the collar adding lemma.
{"title":"Key Facts about Skyscrapers and Decomposition Space Theory","authors":"Mark Powell, Arunima Ray","doi":"10.1093/oso/9780198841319.003.0026","DOIUrl":"https://doi.org/10.1093/oso/9780198841319.003.0026","url":null,"abstract":"‘Key Facts about Skyscrapers and Decomposition Space Theory’ summarizes the input from Parts I and II needed for the remainder of the proof of the disc embedding theorem. Precise references to previous chapters are provided. This enables Part IV to be read independently from the previous parts, provided the reader is willing to accept the facts from Parts I and II summarized in this chapter. The listed facts include the shrinking of mixed, ramified Bing–Whitehead decompositions of the solid torus, the shrinking of null decompositions consisting of recursively starlike-equivalent sets, the ball to ball theorem, the skyscraper embedding theorem, and the collar adding lemma.","PeriodicalId":272723,"journal":{"name":"The Disc Embedding Theorem","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124027894","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-20DOI: 10.1093/oso/9780198841319.003.0002
Arunima Ray
‘Outline of the Upcoming Proof’ provides a comprehensive outline of the proof of the disc embedding theorem. The disc embedding theorem for topological 4-manifolds, due to Michael Freedman, underpins virtually all our understanding of topological 4-manifolds. The famously intricate proof utilizes techniques from both decomposition space theory and smooth manifold topology. The latter is used to construct an infinite iterated object, called a skyscraper, and the former to construct homeomorphisms from a given topological space to a quotient space. The detailed proof of the disc embedding theorem is the core aim of this book. In this chapter, a comprehensive outline of the proof is provided, indicating the chapters in which each aspect is discussed in detail.
{"title":"Outline of the Upcoming Proof","authors":"Arunima Ray","doi":"10.1093/oso/9780198841319.003.0002","DOIUrl":"https://doi.org/10.1093/oso/9780198841319.003.0002","url":null,"abstract":"‘Outline of the Upcoming Proof’ provides a comprehensive outline of the proof of the disc embedding theorem. The disc embedding theorem for topological 4-manifolds, due to Michael Freedman, underpins virtually all our understanding of topological 4-manifolds. The famously intricate proof utilizes techniques from both decomposition space theory and smooth manifold topology. The latter is used to construct an infinite iterated object, called a skyscraper, and the former to construct homeomorphisms from a given topological space to a quotient space. The detailed proof of the disc embedding theorem is the core aim of this book. In this chapter, a comprehensive outline of the proof is provided, indicating the chapters in which each aspect is discussed in detail.","PeriodicalId":272723,"journal":{"name":"The Disc Embedding Theorem","volume":"184 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116064066","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-20DOI: 10.1093/oso/9780198841319.003.0008
Daniel Kasprowski, Min Hoon Kim
Mixed Bing–Whitehead decompositions are a special class of toroidal decompositions of the 3-sphere, defined as the intersection of infinite nested sequences of solid tori. The Bing decomposition and the Whitehead decomposition from previous chapters are both examples of mixed Bing–Whitehead decompositions. In this chapter a precise criterion for when toroidal decompositions shrink is given, in terms of a ‘disc replicating function’. In the case of mixed Bing–Whitehead decomposition, this measures the relative numbers of Bing and Whitehead doubling in the sequence of solid tori in the definition. Mixed Bing–Whitehead decompositions are related to the boundaries of skyscrapers, and the shrinking theorem proved in this chapter will be key to the eventual proof of the disc embedding theorem.
{"title":"Mixed Bing–Whitehead Decompositions","authors":"Daniel Kasprowski, Min Hoon Kim","doi":"10.1093/oso/9780198841319.003.0008","DOIUrl":"https://doi.org/10.1093/oso/9780198841319.003.0008","url":null,"abstract":"Mixed Bing–Whitehead decompositions are a special class of toroidal decompositions of the 3-sphere, defined as the intersection of infinite nested sequences of solid tori. The Bing decomposition and the Whitehead decomposition from previous chapters are both examples of mixed Bing–Whitehead decompositions. In this chapter a precise criterion for when toroidal decompositions shrink is given, in terms of a ‘disc replicating function’. In the case of mixed Bing–Whitehead decomposition, this measures the relative numbers of Bing and Whitehead doubling in the sequence of solid tori in the definition. Mixed Bing–Whitehead decompositions are related to the boundaries of skyscrapers, and the shrinking theorem proved in this chapter will be key to the eventual proof of the disc embedding theorem.","PeriodicalId":272723,"journal":{"name":"The Disc Embedding Theorem","volume":"168 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115697052","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-20DOI: 10.1093/oso/9780198841319.003.0020
P. Orson, Mark Powell, Arunima Ray
The s-cobordism theorem, the sphere embedding theorem, and the Poincaré conjecture comprise three key consequences of the disc embedding theorem. The chapter begins by explaining in detail how to use the disc embedding theorem to prove the 5-dimensional s-cobordism theorem and the sphere embedding theorem. The sphere embedding theorem is the output from the disc embedding theorem that one wants in many situations. A version of the Poincaré conjecture is proven, specifically that every smooth homotopy 4-sphere is homeomorphic to the 4-sphere. All the results proved in this chapter are category losing; that is, they require smooth input but only produce homeomorphisms.
{"title":"The s-cobordism Theorem, the Sphere Embedding Theorem, and the Poincaré Conjecture","authors":"P. Orson, Mark Powell, Arunima Ray","doi":"10.1093/oso/9780198841319.003.0020","DOIUrl":"https://doi.org/10.1093/oso/9780198841319.003.0020","url":null,"abstract":"The s-cobordism theorem, the sphere embedding theorem, and the Poincaré conjecture comprise three key consequences of the disc embedding theorem. The chapter begins by explaining in detail how to use the disc embedding theorem to prove the 5-dimensional s-cobordism theorem and the sphere embedding theorem. The sphere embedding theorem is the output from the disc embedding theorem that one wants in many situations. A version of the Poincaré conjecture is proven, specifically that every smooth homotopy 4-sphere is homeomorphic to the 4-sphere. All the results proved in this chapter are category losing; that is, they require smooth input but only produce homeomorphisms.","PeriodicalId":272723,"journal":{"name":"The Disc Embedding Theorem","volume":"33 17","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"113936001","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-20DOI: 10.1093/oso/9780198841319.003.0006
Stefan Behrens, C. Davis, Mark Powell, Arunima Ray
‘A Decomposition That Does Not Shrink’ gives a nontrivial example of a decomposition of the 3-sphere such that the corresponding quotient space is not homeomorphic to the 3-sphere. The decomposition in question is called the Bing-2 decomposition. Similar to the Bing decomposition from the previous chapter, it consists of the connected components of the intersection of an infinite sequence of nested solid tori. However, unlike the Bing decomposition, the Bing-2 decomposition does not shrink. This indicates the subtlety of the question of which decompositions shrink. The question of when certain decompositions of the 3-sphere shrink is central to the proof of the disc embedding theorem.
{"title":"A Decomposition That Does Not Shrink","authors":"Stefan Behrens, C. Davis, Mark Powell, Arunima Ray","doi":"10.1093/oso/9780198841319.003.0006","DOIUrl":"https://doi.org/10.1093/oso/9780198841319.003.0006","url":null,"abstract":"‘A Decomposition That Does Not Shrink’ gives a nontrivial example of a decomposition of the 3-sphere such that the corresponding quotient space is not homeomorphic to the 3-sphere. The decomposition in question is called the Bing-2 decomposition. Similar to the Bing decomposition from the previous chapter, it consists of the connected components of the intersection of an infinite sequence of nested solid tori. However, unlike the Bing decomposition, the Bing-2 decomposition does not shrink. This indicates the subtlety of the question of which decompositions shrink. The question of when certain decompositions of the 3-sphere shrink is central to the proof of the disc embedding theorem.","PeriodicalId":272723,"journal":{"name":"The Disc Embedding Theorem","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130757212","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-20DOI: 10.1093/oso/9780198841319.003.0014
Stefan Behrens, M. Powell, Arunima Ray
Architecture of Towers and Skyscrapers formalizes the results from the previous chapter, regarding the structure of gropes and towers, and establishes the notation used for towers and skyscrapers in the remainder of the book. In particular, the boundaries of towers and skyscrapers are carefully described. The boundaries are divided into subsets called the floor, the walls, and the ceiling, and the topology of each of them is identified. The walls are associated with certain mixed Bing–Whitehead decompositions from a previous chapter. How the endpoint compactification of a tower corresponds to a quotient space with respect to a decomposition is also described.
{"title":"Architecture of Infinite Towers and Skyscrapers","authors":"Stefan Behrens, M. Powell, Arunima Ray","doi":"10.1093/oso/9780198841319.003.0014","DOIUrl":"https://doi.org/10.1093/oso/9780198841319.003.0014","url":null,"abstract":"Architecture of Towers and Skyscrapers formalizes the results from the previous chapter, regarding the structure of gropes and towers, and establishes the notation used for towers and skyscrapers in the remainder of the book. In particular, the boundaries of towers and skyscrapers are carefully described. The boundaries are divided into subsets called the floor, the walls, and the ceiling, and the topology of each of them is identified. The walls are associated with certain mixed Bing–Whitehead decompositions from a previous chapter. How the endpoint compactification of a tower corresponds to a quotient space with respect to a decomposition is also described.","PeriodicalId":272723,"journal":{"name":"The Disc Embedding Theorem","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131950488","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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