Min Hoon Kim, P. Orson, Junghwan Park, Arunima Ray
{"title":"Good Groups","authors":"Min Hoon Kim, P. Orson, Junghwan Park, Arunima Ray","doi":"10.1093/oso/9780198841319.003.0019","DOIUrl":"https://doi.org/10.1093/oso/9780198841319.003.0019","url":null,"abstract":"Good groups are defined in terms of whether capped gropes of height 1.5 contain certain types of immersed discs. The disc embedding theorem holds for 4-manifolds with good fundamental group. It is proven that the infinite cyclic group and finite groups are good, and that extensions and colimits of good groups are good. This shows that all elementary amenable groups are good. The proofs use grope height raising and contraction, together with an analysis of how fundamental group elements behave under these operations. A central open problem in the study of topological 4-manifolds is to determine precisely which groups are good.","PeriodicalId":272723,"journal":{"name":"The Disc Embedding Theorem","volume":"2 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":"133846595","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}
{"title":"Skyscrapers Are Standard: An Overview","authors":"Stefan Behrens","doi":"10.1093/oso/9780198841319.003.0027","DOIUrl":"https://doi.org/10.1093/oso/9780198841319.003.0027","url":null,"abstract":"‘Skyscrapers Are Standard: An Overview’ provides a detailed outline of the upcoming proof that a skyscraper is homeomorphic to the standard 2-handle, relative to the attaching region. Since the proof is technically challenging, this chapter serves to introduce the key ideas without cumbersome notation. The key points to keep in mind during the proof are enumerated. Briefly, the proof consists of finding a common subset, called the design, of both the given skyscraper and the standard 2-handle. This is accomplished by studying the vertical boundaries of skyscrapers. Next, one would wish to show that the decompositions of the skyscraper and the 2-handle, arising from the connected components of the complements of the common subset, shrink. The decompositions have to be modified to achieve this.","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":"126349283","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}
{"title":"The Alexander Gored Ball and the Bing Decomposition","authors":"Stefan Behrens, Min Hoon Kim","doi":"10.1093/oso/9780198841319.003.0005","DOIUrl":"https://doi.org/10.1093/oso/9780198841319.003.0005","url":null,"abstract":"‘The Alexander Gored Ball and the Bing Decomposition’ provides a concrete and nontrivial application of the tools of decomposition space theory introduced in the previous chapter. The complement of the Alexander horned ball in the 3-sphere is called the Alexander gored ball. This space is described in three distinct ways: as an intersection of 3-balls; as a 3-dimensional grope; and as a decomposition space. Bing’s theorem that the double of the Alexander gored ball is homeomorphic to the 3-sphere is presented. This gives the first example of a truly nontrivial shrink and, moreover, an example of an exotic involution of the 3-sphere.","PeriodicalId":272723,"journal":{"name":"The Disc Embedding Theorem","volume":"39 2 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":"130621729","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}
Stefan Behrens, Allison N. Miller, M. Nagel, P. Teichner
{"title":"The Schoenflies Theorem after Mazur, Morse, and Brown","authors":"Stefan Behrens, Allison N. Miller, M. Nagel, P. Teichner","doi":"10.1093/oso/9780198841319.003.0003","DOIUrl":"https://doi.org/10.1093/oso/9780198841319.003.0003","url":null,"abstract":"‘The Schoenflies Theorem after Mazur, Morse, and Brown’ provides two proofs of the Schoenflies theorem. The Schoenflies theorem states that every bicollared embedding of an (n – 1)-sphere in the n-sphere splits the n-sphere into two balls. This chapter provides two proofs. The first is due to Mazur and Morse; it utilizes an infinite ‘swindle’ and a classical technique called push-pull. The second proof, due to Brown, serves as an introduction to shrinking, or decomposition space theory. The latter is a beautiful, but outmoded, branch of topology that can be used to produce non-differentiable homeomorphisms between manifolds, especially from a manifold to a quotient space. Techniques from decomposition space theory are essential in the proof of the disc embedding theorem.","PeriodicalId":272723,"journal":{"name":"The Disc Embedding Theorem","volume":"110 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":"117218507","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}
{"title":"Grope Height Raising and 1-storey Capped Towers","authors":"P. Feller, Mark D. Powell, Arunima Ray","doi":"10.1093/oso/9780198841319.003.0017","DOIUrl":"https://doi.org/10.1093/oso/9780198841319.003.0017","url":null,"abstract":"‘Grope Height Raising and 1-Storey Capped Towers’ upgrades the capped gropes constructed in the previous chapter to 1-storey capped towers. Grope height raising is a technique that shows that every capped grope of height at least 1.5 can be improved to a capped grope of arbitrary height. The technique is explained in this chapter in detail, and used multiple times in the rest of the proof. The chapter closes by showing how to extend capped gropes to 1-storey capped towers. This crucially uses the hypothesis that the fundamental group is good. It is the single place in the proof of the disc embedding theorem that requires this hypothesis.","PeriodicalId":272723,"journal":{"name":"The Disc Embedding Theorem","volume":"100 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":"131803418","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}
{"title":"Intersection Numbers and the Statement of the Disc Embedding Theorem","authors":"Mark Powell, Arunima Ray","doi":"10.1093/oso/9780198841319.003.0011","DOIUrl":"https://doi.org/10.1093/oso/9780198841319.003.0011","url":null,"abstract":"‘Intersection Numbers and the Statement of the Disc Embedding Theorem’ provides detailed definitions of some of the notions involved in the statement of the disc embedding theorem, focusing specifically on intersection numbers. The chapter begins with a detailed analysis of immersions, regular homotopies, finger moves, and Whitney moves. Then it defines intersection and self-intersection numbers for families of discs and spheres, taking values in the group ring of the fundamental group of the ambient space, with the correct relations. Then it enumerates certain properties of intersection numbers, in particular relating them to the existence of Whitney discs. This work enables the disc embedding theorem to be stated carefully.","PeriodicalId":272723,"journal":{"name":"The Disc Embedding Theorem","volume":"215 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":"130866795","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}
{"title":"Replicable Rooms and Boundary Shrinkable Skyscrapers","authors":"Stefan Behrens, Mark Powell, Arunima Ray","doi":"10.1093/oso/9780198841319.003.0024","DOIUrl":"https://doi.org/10.1093/oso/9780198841319.003.0024","url":null,"abstract":"‘Replicable Rooms and Boundary Shrinkable Skyscrapers’ points out precisely which properties of skyscrapers are required in the remainder of the proof of the disc embedding theorem. To achieve this, it introduces an abstraction of towers, known as buildings. The required properties for a generalized skyscraper include boundary shrinkability and replicability. The former allows the conclusion that the vertical boundary of a generalized skyscraper is a solid torus. Replicability ensures that any generalized skyscraper contains uncountably many other skyscrapers as subsets. Both of the above properties will be essential in the construction of the design in a subsequent chapter.","PeriodicalId":272723,"journal":{"name":"The Disc Embedding Theorem","volume":"271 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":"115984032","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}
{"title":"Gropes, Towers, and Skyscrapers","authors":"Mark Powell, Arunima Ray","doi":"10.1093/oso/9780198841319.003.0012","DOIUrl":"https://doi.org/10.1093/oso/9780198841319.003.0012","url":null,"abstract":"Gropes, towers, and skyscrapers are carefully defined. These are the objects that the rest of Part II studies and seeks to construct. All three are 4-manifolds with boundary, obtained from stacking thickened surfaces on top of one another. Gropes are constructed from thickened orientable surfaces with positive genus, each stage attached to a symplectic basis of curves for the homology of the previous stage. Towers have an additional type of stage obtained from plumbed thickened discs. A skyscraper is the endpoint compactification of an infinite tower. An introduction to endpoint compactifications is included. The notion of a good group is also defined.","PeriodicalId":272723,"journal":{"name":"The Disc Embedding Theorem","volume":"11 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":"126722967","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}
{"title":"The Development of Topological 4-manifold Theory","authors":"Mark Powell, Arunima Ray","doi":"10.1093/oso/9780198841319.003.0021","DOIUrl":"https://doi.org/10.1093/oso/9780198841319.003.0021","url":null,"abstract":"The development of topological 4-manifold theory is described in the form of a flowchart showing the interdependence among many key statements in the theory. In particular, the flowchart demonstrates how the theory crucially relies on the constructions in this book, what goes into the work of Quinn on smoothing, normal bundles, and transversality, and what is needed to deduce the famous consequences, such as the classification of closed, simply connected, topological 4-manifolds, the category preserving Poincaré conjecture, and the existence of exotic smooth structures on 4-dimensional Euclidean space. Precise statements of the results, brief indications of some proofs, and extensive references are provided.","PeriodicalId":272723,"journal":{"name":"The Disc Embedding Theorem","volume":"40 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":"127411491","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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