Pub Date : 2021-07-20DOI: 10.1093/oso/9780198841319.003.0023
Min Hoon Kim, P. Orson, Junghwan Park, Arunima Ray
Open problems in the study of topological 4-manifolds are explained in detail. An important open problem is to determine whether the disc embedding theorem and its antecedents hold for all groups; in other words, whether all groups are good. The disc embedding conjecture and the surgery conjecture are stated. The relationships between these conjectures and their various reformulations are explained. Of particular interest are the reformulations in terms of freely slicing certain infinite families of links. In particular, the surgery conjecture is true if and only if all good boundary links are freely slice. Good boundary links are the many-component analogues of Alexander polynomial one knots.
{"title":"Open Problems","authors":"Min Hoon Kim, P. Orson, Junghwan Park, Arunima Ray","doi":"10.1093/oso/9780198841319.003.0023","DOIUrl":"https://doi.org/10.1093/oso/9780198841319.003.0023","url":null,"abstract":"Open problems in the study of topological 4-manifolds are explained in detail. An important open problem is to determine whether the disc embedding theorem and its antecedents hold for all groups; in other words, whether all groups are good. The disc embedding conjecture and the surgery conjecture are stated. The relationships between these conjectures and their various reformulations are explained. Of particular interest are the reformulations in terms of freely slicing certain infinite families of links. In particular, the surgery conjecture is true if and only if all good boundary links are freely slice. Good boundary links are the many-component analogues of Alexander polynomial one knots.","PeriodicalId":272723,"journal":{"name":"The Disc Embedding Theorem","volume":"5 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":"127386475","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/9780198841319.003.0025
Daniel Kasprowski, Mark Powell, Arunima Ray
The collar adding lemma is a key ingredient in the proof of the disc embedding theorem. Specifically, it proves that a skyscraper with an added collar is homeomorphic to the standard 4-dimensional 2-handle. The proof is similar to the proof in a previous chapter that the Alexander gored ball with an added collar is homeomorphic to the standard 3-ball. Roughly speaking, a skyscraper is seen as the quotient space of the 4-ball corresponding to a certain decomposition. The added collar allows the decomposition to be modified so that the resulting decomposition shrinks; that is, the corresponding quotient space, which is identified with the skyscraper with an added collar, is homeomorphic to the original 4-ball.
{"title":"The Collar Adding Lemma","authors":"Daniel Kasprowski, Mark Powell, Arunima Ray","doi":"10.1093/oso/9780198841319.003.0025","DOIUrl":"https://doi.org/10.1093/oso/9780198841319.003.0025","url":null,"abstract":"The collar adding lemma is a key ingredient in the proof of the disc embedding theorem. Specifically, it proves that a skyscraper with an added collar is homeomorphic to the standard 4-dimensional 2-handle. The proof is similar to the proof in a previous chapter that the Alexander gored ball with an added collar is homeomorphic to the standard 3-ball. Roughly speaking, a skyscraper is seen as the quotient space of the 4-ball corresponding to a certain decomposition. The added collar allows the decomposition to be modified so that the resulting decomposition shrinks; that is, the corresponding quotient space, which is identified with the skyscraper with an added collar, is homeomorphic to the original 4-ball.","PeriodicalId":272723,"journal":{"name":"The Disc Embedding Theorem","volume":"61 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":"121709160","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/9780198841319.003.0016
Wojciech Politarczyk, Mark Powell, Arunima Ray
‘From Immersed Discs to Capped Gropes’ begins the proof of the disc embedding theorem in earnest. Starting with the immersed discs provided by the hypotheses of the disc embedding theorem, capped gropes with the same boundary and with suitable dual gropes are produced. This uses a sequence of the geometric moves introduced in the previous chapter. The two propositions in this chapter are technical, but vital. In subsequent chapters, the capped gropes will be upgraded to capped towers, and then to skyscrapers. The final step of the proof will consist of showing that skyscrapers are homeomorphic to the standard 2-handle, relative to the attaching region.
{"title":"From Immersed Discs to Capped Gropes","authors":"Wojciech Politarczyk, Mark Powell, Arunima Ray","doi":"10.1093/oso/9780198841319.003.0016","DOIUrl":"https://doi.org/10.1093/oso/9780198841319.003.0016","url":null,"abstract":"‘From Immersed Discs to Capped Gropes’ begins the proof of the disc embedding theorem in earnest. Starting with the immersed discs provided by the hypotheses of the disc embedding theorem, capped gropes with the same boundary and with suitable dual gropes are produced. This uses a sequence of the geometric moves introduced in the previous chapter. The two propositions in this chapter are technical, but vital. In subsequent chapters, the capped gropes will be upgraded to capped towers, and then to skyscrapers. The final step of the proof will consist of showing that skyscrapers are homeomorphic to the standard 2-handle, relative to the attaching region.","PeriodicalId":272723,"journal":{"name":"The Disc Embedding Theorem","volume":"6 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":"124960535","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/9780198841319.003.0009
J. Meier, P. Orson, Arunima Ray
‘Shrinking Starlike Sets’ presents a proof that null decompositions with recursively starlike-equivalent elements shrink. Unlike in previous chapters, the shrink is produced abstractly rather than explicitly. The chapter proceeds in steps, first establishing the shrinking of null decompositions with starlike elements, then of those with starlike-equivalent elements, and then finally of those with recursively starlike-equivalent elements. In the eventual proof of the disc embedding theorem, a decomposition of the 4-ball consisting of recursively starlike-equivalent elements will be produced. These consist of the ‘holes’, along with certain ‘red blood cell discs’. The result from this chapter will be used to show that the decomposition shrinks; this is called the α-shrink.
{"title":"Shrinking Starlike Sets","authors":"J. Meier, P. Orson, Arunima Ray","doi":"10.1093/oso/9780198841319.003.0009","DOIUrl":"https://doi.org/10.1093/oso/9780198841319.003.0009","url":null,"abstract":"‘Shrinking Starlike Sets’ presents a proof that null decompositions with recursively starlike-equivalent elements shrink. Unlike in previous chapters, the shrink is produced abstractly rather than explicitly. The chapter proceeds in steps, first establishing the shrinking of null decompositions with starlike elements, then of those with starlike-equivalent elements, and then finally of those with recursively starlike-equivalent elements. In the eventual proof of the disc embedding theorem, a decomposition of the 4-ball consisting of recursively starlike-equivalent elements will be produced. These consist of the ‘holes’, along with certain ‘red blood cell discs’. The result from this chapter will be used to show that the decomposition shrinks; this is called the α-shrink.","PeriodicalId":272723,"journal":{"name":"The Disc Embedding Theorem","volume":"57 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":"128233112","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/9780198841319.003.0029
M. Freedman
This is not a proof of the Poincaré conjecture, but a discussion of the proof, its context, and some of the people who played a prominent role. It is a personal, anecdotal account. There may be omissions or transpositions, as these recollections are 40 years old and not supported by contemporaneous notes, but memories feel surprisingly fresh....
{"title":"Afterword: PC4 at Age 40","authors":"M. Freedman","doi":"10.1093/oso/9780198841319.003.0029","DOIUrl":"https://doi.org/10.1093/oso/9780198841319.003.0029","url":null,"abstract":"This is not a proof of the Poincaré conjecture, but a discussion of the proof, its context, and some of the people who played a prominent role. It is a personal, anecdotal account. There may be omissions or transpositions, as these recollections are 40 years old and not supported by contemporaneous notes, but memories feel surprisingly fresh....","PeriodicalId":272723,"journal":{"name":"The Disc Embedding Theorem","volume":"58 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":"131224407","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/9780198841319.003.0004
C. Davis, B. Kalmár, Min Hoon Kim, H. Rüping
‘Decomposition Space Theory and the Bing Shrinking Criterion’ gives a proof of the central Bing shrinking criterion and then provides an introduction to the key notions of the field of decomposition space theory. The chapter begins by proving the Bing shrinking criterion, which characterizes when a given map between compact metric spaces is approximable by homeomorphisms. Next, it develops the elements of the theory of decomposition spaces. A key fact is that a decomposition space associated with an upper semi-continuous decomposition of a compact metric space is again a compact metric space. Decomposition spaces are key in the proof of the disc embedding theorem.
{"title":"Decomposition Space Theory and the Bing Shrinking Criterion","authors":"C. Davis, B. Kalmár, Min Hoon Kim, H. Rüping","doi":"10.1093/oso/9780198841319.003.0004","DOIUrl":"https://doi.org/10.1093/oso/9780198841319.003.0004","url":null,"abstract":"‘Decomposition Space Theory and the Bing Shrinking Criterion’ gives a proof of the central Bing shrinking criterion and then provides an introduction to the key notions of the field of decomposition space theory. The chapter begins by proving the Bing shrinking criterion, which characterizes when a given map between compact metric spaces is approximable by homeomorphisms. Next, it develops the elements of the theory of decomposition spaces. A key fact is that a decomposition space associated with an upper semi-continuous decomposition of a compact metric space is again a compact metric space. Decomposition spaces are key in the proof of the disc embedding theorem.","PeriodicalId":272723,"journal":{"name":"The Disc Embedding Theorem","volume":"212 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":"133527286","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/9780198841319.003.0001
Stefan Behrens, Mark Powell, Arunima Ray
‘Context for the Disc Embedding Theorem’ explains why the theorem is the central result in the study of topological 4-manifolds. After recalling surgery theory and the proof of the s-cobordism theorem for high-dimensional manifolds, the chapter explains what goes wrong when trying to apply the same techniques in four dimensions and how to start overcoming these problems. The complete statement of the disc embedding theorem is provided. Finally the most important consequences to manifold theory are listed, including a proof of why Alexander polynomial one knots are topologically slice and the existence of exotic smooth structures on 4-dimensional Euclidean space.
{"title":"Context for the Disc Embedding Theorem","authors":"Stefan Behrens, Mark Powell, Arunima Ray","doi":"10.1093/oso/9780198841319.003.0001","DOIUrl":"https://doi.org/10.1093/oso/9780198841319.003.0001","url":null,"abstract":"‘Context for the Disc Embedding Theorem’ explains why the theorem is the central result in the study of topological 4-manifolds. After recalling surgery theory and the proof of the s-cobordism theorem for high-dimensional manifolds, the chapter explains what goes wrong when trying to apply the same techniques in four dimensions and how to start overcoming these problems. The complete statement of the disc embedding theorem is provided. Finally the most important consequences to manifold theory are listed, including a proof of why Alexander polynomial one knots are topologically slice and the existence of exotic smooth structures on 4-dimensional Euclidean space.","PeriodicalId":272723,"journal":{"name":"The Disc Embedding Theorem","volume":"234 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":"124599680","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/9780198841319.003.0013
D. McCoy, Junghwan Park, Arunima Ray
‘Picture Camp’ provides a review of Kirby handle calculus for describing 4-manifolds via decorated link diagrams, as well as techniques for how to simplify such diagrams. This chapter applies these techniques to describe gropes and towers, from the previous chapter, using Kirby diagrams. In addition to decorated links, the diagrams include the information of framings for the attaching and tip regions. In particular, it is shown how to combine two diagrams together when the corresponding spaces are identified along their attaching and tip regions. The chapter also relates the combinatorics of gropes and towers to the combinatorics of the associated link diagrams.
{"title":"Picture Camp","authors":"D. McCoy, Junghwan Park, Arunima Ray","doi":"10.1093/oso/9780198841319.003.0013","DOIUrl":"https://doi.org/10.1093/oso/9780198841319.003.0013","url":null,"abstract":"‘Picture Camp’ provides a review of Kirby handle calculus for describing 4-manifolds via decorated link diagrams, as well as techniques for how to simplify such diagrams. This chapter applies these techniques to describe gropes and towers, from the previous chapter, using Kirby diagrams. In addition to decorated links, the diagrams include the information of framings for the attaching and tip regions. In particular, it is shown how to combine two diagrams together when the corresponding spaces are identified along their attaching and tip regions. The chapter also relates the combinatorics of gropes and towers to the combinatorics of the associated link diagrams.","PeriodicalId":272723,"journal":{"name":"The Disc Embedding Theorem","volume":"80 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":"115593430","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/9780198841319.003.0022
P. Orson, Mark Powell, Arunima Ray
Surgery theory and the classification of simply connected 4-manifolds comprise two key consequences of the disc embedding theorem. The chapter begins with an introduction to surgery theory from the perspective of 4-manifolds. In particular, the terms and maps in the surgery sequence are defined, and an explanation is given as to how the sphere embedding theorem, with the added ingredient of topological transversality, can be used to define the maps in the surgery sequence and show that it is exact. The surgery sequence is applied to classify simply connected closed 4-manifolds up to homeomorphism. The chapter closes with a survey of related classification results.
{"title":"Surgery Theory and the Classification of Closed, Simply Connected 4-manifolds","authors":"P. Orson, Mark Powell, Arunima Ray","doi":"10.1093/oso/9780198841319.003.0022","DOIUrl":"https://doi.org/10.1093/oso/9780198841319.003.0022","url":null,"abstract":"Surgery theory and the classification of simply connected 4-manifolds comprise two key consequences of the disc embedding theorem. The chapter begins with an introduction to surgery theory from the perspective of 4-manifolds. In particular, the terms and maps in the surgery sequence are defined, and an explanation is given as to how the sphere embedding theorem, with the added ingredient of topological transversality, can be used to define the maps in the surgery sequence and show that it is exact. The surgery sequence is applied to classify simply connected closed 4-manifolds up to homeomorphism. The chapter closes with a survey of related classification results.","PeriodicalId":272723,"journal":{"name":"The Disc Embedding Theorem","volume":"24 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":"117133143","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/9780198841319.003.0018
Allison N. Miller, Mark Powell, Arunima Ray
‘Tower Height Raising and Embedding’ shows how to raise the height of towers, as well as how to detect embedded towers within a given tower. Raising the number of storeys of a capped tower by one is a construction similar in spirit to grope height raising, but more sophisticated. The new aspect, which receives careful treatment, is the geometric control needed to make the top storey arbitrarily small. An n-storey capped tower contains a capped tower with (n + 1) storeys and the same attaching region, and this can be realized by an embedding that places connected components of the top storey into balls of arbitrarily chosen small diameter. Consequently, endpoint compactifications of infinite towers may be embedded as well.
{"title":"Tower Height Raising and Embedding","authors":"Allison N. Miller, Mark Powell, Arunima Ray","doi":"10.1093/oso/9780198841319.003.0018","DOIUrl":"https://doi.org/10.1093/oso/9780198841319.003.0018","url":null,"abstract":"‘Tower Height Raising and Embedding’ shows how to raise the height of towers, as well as how to detect embedded towers within a given tower. Raising the number of storeys of a capped tower by one is a construction similar in spirit to grope height raising, but more sophisticated. The new aspect, which receives careful treatment, is the geometric control needed to make the top storey arbitrarily small. An n-storey capped tower contains a capped tower with (n + 1) storeys and the same attaching region, and this can be realized by an embedding that places connected components of the top storey into balls of arbitrarily chosen small diameter. Consequently, endpoint compactifications of infinite towers may be embedded as well.","PeriodicalId":272723,"journal":{"name":"The Disc Embedding Theorem","volume":"17 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":"124502378","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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