Induced mappings on $C_n(X)/{C_n}_K(X)$

Given a continuum $X$ and $n\in\mathbb{N}$. Let $C_n(X)$ be the hyperspace of all nonempty closed subsets of $X$ with at most $n$ components. Let ${C_n}_K(X)$ be the hyperspace of all elements in $C_n(X)$ containing $K$ where $K$ is a compact subset of $X$. $C^n_K(X)$ denotes the quotient space $C_n(X)/{C_n}_K(X)$. Given a mapping $f:X\to Y$ between continua, let $C_n(f):C_n(X)\to C_n(Y)$ be the induced mapping by $f$, defined by $C_n(f)(A)=f(A)$. We denote the natural induced mapping between $C^n_K(X)$ and $C^n_{f(K)}(Y)$ by $C^n_K(f)$. In this paper, we study relationships among the mappings $f$, $C_n(f)$ and $C^n_K(f)$ for the following classes of mappings: almost monotone, atriodic, confluent, joining, light, monotone, open, OM, pseudo-confluent, quasi-monotone, semi-confluent, strongly freely decomposable, weakly confluent, and weakly monotone.

The Collar Adding Lemma

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 Schoenflies Theorem after Mazur, Morse, and Brown

‘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.

Outline of the Upcoming Proof

‘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

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.

A Decomposition That Does Not Shrink

‘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.

Covariant functions of characters of compact subgroups

This paper presents a systematic study for abstract harmonic analysis on classical Banach spaces of covariant functions of characters of compact subgroups. Let G be a locally compact group and H be a compact subgroup of G. Suppose that $$\xi :H\rightarrow \mathbb {T}$$ ξ : H → T is a character, $$1\le p<\infty$$ 1 ≤ p < ∞ and $$L_\xi ^p(G,H)$$ L ξ p ( G , H ) is the set of all covariant functions of $$\xi$$ ξ in $$L^p(G)$$ L p ( G ) . It is shown that $$L^p_\xi (G,H)$$ L ξ p ( G , H ) is isometrically isomorphic to a quotient space of $$L^p(G)$$ L p ( G ) . It is also proven that $$L^q_\xi (G,H)$$ L ξ q ( G , H ) is isometrically isomorphic to the dual space $$L^p_\xi (G,H)^*$$ L ξ p ( G , H ) ∗ , where q is the conjugate exponent of p. The paper is concluded by some results for the case that G is compact.