# Topology of 4-Manifolds (PMS-39)

Michael H. Freedman
Frank Quinn
Pages: 268
https://www.jstor.org/stable/j.ctt7ztz2d

1. Front Matter
(pp. i-iv)
(pp. v-2)
3. Introduction
(pp. 3-8)

The study of the topology of manifolds has turned out to be dependent on dimension in a curious way. Manifolds of dimension 2 are a classical subject, and have been largely understood for half a century. Historically the next major progress was in dimension 3; Moise in 1952 reduced the topological theory to the piecewise linear (or equivalently smooth) category, and Papakyriakopoulos in 1957 established the beginnings of embedded surface theory. But then dimension 4 was skipped over, and higher dimensions (≥ 5) were developed next. The two key events were the development of the methods of smooth and PL...

4. ### Part I. Embeddings of Disks

• CHAPTER 1 Basic tools
(pp. 11-29)

In this chapter the principal methods for manipulation of immersions of surfaces are described. The raw material is provided by the immersion lemma 1.2, which is assumed as an axiom in Part I. This and the twisting operation of 1.3 are used to construct (immersed) Whitney disks in 1.4. These disks are used to define Whitney moves, which are one of the principal geometric operations in topology of any dimension. Finger moves and connected sums are developed in sections 1.5 and 1.8 as ways to remove intersections, especially with Whitney disks. Intersection numbers, described in 1.7, give algebraic conditions under...

• CHAPTER 2 Capped gropes
(pp. 30-47)

Capped gropes are 2-complexes with the same regular neighborhood as a fixed surface, usually a disk. Intersections which are “more subtle” than the standard plumbing points can be defined in terms of this more complicated description of the neighborhood.

The first three sections describe capped surfaces, and this construction is iterated in 2.4 to define capped gropes. The key property of gropes is that once a certain height has been obtained they can “grow.” This is proved in 2.7, and then exploited in the remainder of the chapter. It is used to disengage the image of gropes from the fundamental...

• CHAPTER 3 Capped towers
(pp. 48-61)

A capped grope is built of layers of surfaces, and a layer of disks (the caps) at the top. We will need more complicated objects, “towers”, whose building blocks are themselves capped gropes. These are defined in the first two sections, and constructed from more primitive data in 3.3. The height of a tower is raised, finitely many times in 3.5 and infinitely in 3.8. The convergent infinite towers of 3.8 provide the raw material for the decomposition theory constructions of the next chapter.

The original proof used somewhat simpler towers, using only accessory disks. Towers of gropes are used...

• CHAPTER 4 Parameterization of convergent towers
(pp. 62-84)

It follows from section 3.8 that when the hypotheses of the main theorem are satisfied then the manifold has a subset homeomorphic to a certain subset ofD4, namely a pinched regular neighborhood of a convergent infinite tower. The objective of this chapter is to show that this pinched neighborhood contains a correctly framed embedded disk.

We establish some notation to be used throughout this chapter.CD2D2is a pinched regular neighborhood of a disk-like convergent tower, as described in 3.8.C(D2D2) is arranged to beS1D2, and is denoted0C.1C...

• CHAPTER 5 The Embedding Theorems
(pp. 85-91)

The keys to the study of 4-manifolds are embedding theorems for 2-disks, as explained in the introduction. In this chapter precise statements of the embedding results are given, and proofs assembled from the rest of Part I. In all cases the goal is to find a collection of disjointly embedded disks. Some statements can be extended to other surfaces, but because of their crucial importance we focus on disks.

The technically important results are the poly-(finite or cyclic) embeddings of section 5.1, and the controlled embeddings of 5.4. The results of 5.2 and 5.3, in which hypotheses are imposed on...

• CHAPTER 6 Embedding up to s-Cobordism
(pp. 92-98)

In this chapter we prove theorem 5.3, which asserts that a π1-null collection of immersed disks and algebraically transverse spheres, with algebraically trivial intersections, is s-cobordant to an embedding. Some comments are given in 6.4 to explain how the proof breaks down without the π1-null hypothesis.

The model for atransverse pairis two copies ofS2D2plumbed together at one point. Note each sphere is a transverse sphere for the other. This model is also a neighborhood of the pair of spheresS2✕ {pt} ∪ {pt} ✕S2S2S2, which demonstrates that the...

5. ### Part II. Applications to the Structure of Manifolds

• Introduction
(pp. 99-100)

The remainder of this book is devoted to consequences of the embedding theorem. Chapter topics are described briefly in the introduction, and in detail at the beginnings of the chapters. Here we introduce the term “good” for fundamental groups, and comment on the logic and significance of the material.

In Part II “good” is used to refer to fundamental groups for which the embedding theorem is known. In these terms the main theorem asserts that poly-(finite or cyclic) groups are “good.” The intent is to allow easy identification of constraints imposed by the 4-dimensional topology, as distinct for example from...

• CHAPTER 7 h-Cobordisms
(pp. 101-113)

The basic result is that certain 5-dimensional s-cobordisms are products. Section 7.1 concerns the classical case of compact h-cobordisms, while 7.2 and 7.3 deal with controlled and proper h-cobordisms respectively. There are refinements which give more information on the singularities of the product structures. These are used in the next chapter, in studying smooth and handlebody structures, while later chapters will primarily use the basic product result.

Recall that anh-cobordism(rel boundary) is a manifoldWwith∂W=M0M1a union of submanifolds, such thatWdeformation retracts to eachMi. A compact h-cobordism has a...

• CHAPTER 8 Smooth structures
(pp. 114-133)

The main results of this chapter are partial existence and uniqueness results for smooth structures on 4-manifolds. These are limited by smooth nonexistence results obtained from differential geometry, so in fact their main use (in the next Chapter) is to obtain further information about the topological category. We have included discussions of the 4-dimensional smooth nonexistence results, and—to make the idiosyncrasies of this dimension clear—a description of the smoothing theory in other dimensions. In 8.6 and 8.7 appropriate sum-stable and bundle versions are shown to follow the high dimensional pattern. Finally in 8.8 smooth structures in the complement...

• CHAPTER 9 Handlebodies, Normal Bundles, and Transversality
(pp. 134-160)

The principal conclusions of this chapter are that 5-dimensional topological manifolds have handlebody structures; that submanifolds of 4-manifolds have normal bundles; and that such submanifolds can be made transverse by isotopy. Roughly these mean the basic tools of manifold topology are available for use in 4-manifolds, and we can proceed as in other dimensions.

These results also tie up some loose ends. Transversality implies the topological version of the immersion lemma 1.2. Part I is contingent only on this lemma, so the disk embedding theorems apply to topological manifolds. The technical forms of the h-cobordism theorems (7.1D, 7.2C) require both...

• CHAPTER 10 Classifications and Embeddings
(pp. 161-194)

Technically, the main result of the section is the π1-negligible embedding theorem 10.6, which gives existence and uniqueness criteria for codimension zero embeddings of 4-manifolds with boundary. From this we deduce a criterion for a manifold to decompose as a connected sum. Perhaps the most basic result, the classification of closed simply-connected 4-manifolds, is obtained as a corollary of the sum theorem.

These results are presented in the reverse of the logical order because the more general ones are more complicated and technical. The chapter begins, then, with the classification theorem in 10.1. The invariants used in the classification are...

• CHAPTER 11 Surgery
(pp. 195-231)

“Surgery” refers to a collection of techniques for manipulating manifolds. The brief conclusion is that “surgery works” for topological 4-manifolds, provided the fundamental groups are good. In this chapter we survey some of the main developments with interesting 4-dimensional applications, but do not attempt a comprehensive treatment.

The chapter begins with the “plus construction,” which illustrates the basic technique without a lot of machinery. This is used to construct group actions on contractible 4-manifolds in 11.1C. The surgery sequence is described and the new cases of exactness are proved in 11.3. The “surgery program” for classifying manifolds using this sequence...

• CHAPTER 12 Links, and Reformulations of the Embedding Problem
(pp. 232-248)

The embedding problem concerns the necessity of fundamental group restrictions in the disk embedding theorem. The reformulations given in this chapter were developed to explore the problem; either suggest new approaches or bring invariants to bear which would detect counterexamples.

The first section restates the problem in terms of embedding disks in neighborhoods of properly immersed capped gropes. These neighborhoods (when orientable) occur as subsets ofD4; the second section describes these subsets explicitly as complements of certain link slices. The embedding problem is therefore equivalent to a link slice problem.

The third section concerns slices for links inS3....

6. References
(pp. 249-256)
7. Index of Notation
(pp. 257-257)
8. Index of Terminology
(pp. 257-259)