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.

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 Disc Embedding Theorem

2021 ◽  
Keyword(s):  
Graduate Students ◽  
Euclidean Space ◽  
Iterative Process ◽  
Embedding Theorem ◽  
Space Theory ◽  
Topological Category ◽  
Surgery Theory ◽  
Smooth Structures ◽  
Decomposition Space ◽  
Exotic Smooth Structures

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.

Decomposition Space Theory and the Bing Shrinking Criterion

2021 ◽  
pp. 62-76
Author(s):  
Christopher W. Davis ◽  
Boldizsár Kalmár ◽  
Min Hoon Kim ◽  
Henrik Rüping
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Embedding Theorem ◽  
Space Theory ◽  
Compact Metric Space ◽  
Decomposition Spaces ◽  
Compact Metric Spaces ◽  
Space Decomposition ◽  
Decomposition Space ◽  
Continuous Decomposition

‘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.

Skyscrapers Are Standard: The Details

2021 ◽  
pp. 407-446
Author(s):  
Stefan Behrens ◽  
Daniel Kasprowski ◽  
Mark Powell ◽  
Arunima Ray
Keyword(s):  
Connected Components ◽  
Space Theory ◽  
Decomposition Spaces ◽  
Detailed Proof ◽  
Decomposition Space ◽  
Quotient Maps

‘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.

Key Facts about Skyscrapers and Decomposition Space Theory

2021 ◽  
pp. 395-398
Author(s):  
Mark Powell ◽  
Arunima Ray
Keyword(s):  
Solid Torus ◽  
Embedding Theorem ◽  
Space Theory ◽  
Decomposition Space

‘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.

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 Alexander Gored Ball and the Bing Decomposition

2021 ◽  
pp. 76-86
Author(s):  
Stefan Behrens ◽  
Min Hoon Kim
Keyword(s):  
Space Theory ◽  
3 Dimensional ◽  
Decomposition Space

‘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.

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