A Decomposition That Does Not Shrink

2021 ◽  
pp. 87-94
Author(s):  
Stefan Behrens ◽  
Christopher W. Davis ◽  
Mark Powell ◽  
Arunima Ray
Keyword(s):  
Quotient Space ◽  
Infinite Sequence ◽  
Embedding Theorem ◽  
Connected Components

‘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.

The Collar Adding Lemma

2021 ◽  
pp. 391-394
Author(s):  
Daniel Kasprowski ◽  
Mark Powell ◽  
Arunima Ray
Keyword(s):  
Quotient Space ◽  
Embedding Theorem

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.

The Ball to Ball Theorem

2021 ◽  
pp. 131-152
Author(s):  
Stefan Behrens ◽  
Boldizsár Kalmár ◽  
Daniele Zuddas
Keyword(s):  
Embedding Theorem ◽  
Dense Image

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.

The Schoenflies Theorem after Mazur, Morse, and Brown

2021 ◽  
pp. 44-62
Author(s):  
Stefan Behrens ◽  
Allison N. Miller ◽  
Matthias Nagel ◽  
Peter Teichner
Keyword(s):  
Quotient Space ◽  
Embedding Theorem ◽  
Space Theory ◽  
Decomposition Space ◽  
Classical Technique

‘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.

The Whitehead Decomposition

2021 ◽  
pp. 95-102
Author(s):  
Xiaoyi Cui ◽  
Boldizsár Kalmár ◽  
Patrick Orson ◽  
Nathan Sunukjian
Keyword(s):  
Euclidean Space ◽  
Real Line ◽  
Embedding Theorem ◽  
The Real ◽  
Subsequent Chapter ◽  
Simply Connected

‘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.

Outline of the Upcoming Proof

2021 ◽  
pp. 27-42
Author(s):  
Arunima Ray
Keyword(s):  
Topological Space ◽  
Smooth Manifold ◽  
Quotient Space ◽  
Embedding Theorem ◽  
Space Theory ◽  
Detailed Proof ◽  
The Core ◽  
Decomposition Space

‘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.

Architecture of Infinite Towers and Skyscrapers

2021 ◽  
pp. 211-216
Author(s):  
Stefan Behrens ◽  
Mark Powell ◽  
Arunima Ray
Keyword(s):  
Quotient Space

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.

From Immersed Discs to Capped Gropes

2021 ◽  
pp. 227-238
Author(s):  
Wojciech Politarczyk ◽  
Mark Powell ◽  
Arunima Ray
Keyword(s):  
Embedding Theorem

‘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.

Grope Height Raising and 1-storey Capped Towers

2021 ◽  
pp. 239-252
Author(s):  
Peter Feller ◽  
Mark Powell ◽  
Arunima Ray
Keyword(s):  
Fundamental Group ◽  
Embedding Theorem

‘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.

Somethin’ about Scandinavia: Danish Shorts on the Post-war International Scene

2018 ◽  
Author(s):  
C. Claire Thomson
Keyword(s):  
United Nations ◽  
International Collaboration ◽  
Marshall Plan ◽  
The Third ◽  
Animated Film ◽  
International Audience ◽  
Post War ◽  
Nordic Cooperation

Building on the picture of post-war Anglo-Danish documentary collaboration established in the previous chapter, this chapter examines three cases of international collaboration in which Dansk Kulturfilm and Ministeriernes Filmudvalg were involved in the late 1940s and 1950s. They Guide You Across (Ingolf Boisen, 1949) was commissioned to showcase Scandinavian cooperation in the realm of aviation (SAS) and was adopted by the newly-established United Nations Film Board. The complexities of this film’s production, funding and distribution are illustrative of the activities of the UN Film Board in its first years of operation. The second case study considers Alle mine Skibe (All My Ships, Theodor Christensen, 1951) as an example of a film commissioned and funded under the auspices of the Marshall Plan. This US initiative sponsored informational films across Europe, emphasising national solutions to post-war reconstruction. The third case study, Bent Barfod’s animated film Noget om Norden (Somethin’ about Scandinavia, 1956) explains Nordic cooperation for an international audience, but ironically exposed some gaps in inter-Nordic collaboration in the realm of film.

Capitalising upon Globalisation

2018 ◽  
Author(s):  
David M. Webber
Keyword(s):  
Political Economy ◽  
Foreign Policy ◽  
International Development ◽  
Macroeconomic Policy ◽  
Domestic Policy ◽  
Policy Decisions ◽  
Gordon Brown ◽  
Development Policies ◽  
At Home

Having mapped out in the previous chapter, New Labour’s often contradictory and even ‘politically-convenient’ understanding of globalisation, chapter 3 offers analysis of three key areas of domestic policy that Gordon Brown would later transpose to the realm of international development: (i) macroeconomic policy, (ii) business, and (iii) welfare. Since, according to Brown at least, globalisation had resulted in a blurring of the previously distinct spheres of domestic and foreign policy, it made sense for those strategies and policy decisions designed for consumption at home to be transposed abroad. The focus of this chapter is the design of these three areas of domestic policy; the unmistakeable imprint of Brown in these areas and their place in building of New Labour’s political economy. Strikingly, Brown’s hand in these policies and the themes that underpinned them would again reappear in the international development policies explored in much greater detail later in the book.

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