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Four-Dimensional Topology: an Introduction

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Four-Dimensional Topology: an Introduction BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 2, Number 1, January 1980 FOUR-DIMENSIONAL TOPOLOGY: AN INTRODUCTION BY RICHARD MANDELBAUM1 Introduction. This paper is an elaboration of the talk I gave at the American Mathematical Society meeting held at Memphis State University in November, 1977. In that talk I tried to illustrate what was special about four-dimensional topology and mention some of the advances in that subject. I have expanded upon that talk somewhat in this paper in order to give a clearer picture of the geometric ideas and methods currently used in studying four-dimensional manifolds. The basic problem of geometry is the characterization of spaces by means of algebraic invariants. In its simplest form it is the "Side-Angle-Side" theorem of high school geometry which characterizes triangles up to con­ gruence or the 'Angle-Angle' theorem which characterizes them up to similar­ ity. In the more rarefied realm of algebraic and geometric topology we are especially interested in attaching algebraic objects such as groups, rings, modules, etc., to a space M already known to be a compact connected manifold in such a way that if two manifolds M9 M' have the same objects associated to them then they are isomorphic (where isomorphic might mean 'homeomorphic' or 'piecewise-linearly homeomorphic' or 'diffeomorphic'). As a result of the undecidability of the word problem for finitely-presented groups [Bo], [BHP] there is, in general, no finite procedure for deciding in all cases whether two groups given by finite sets of generators and relations are isomorphic. Furthermore, as is well known, any finitely-presented group can be realized as the fundamental group of a compact n-manifold if n > 4. Thus provided n > 4 we have no generally effective procedure for determining if Mi is isomorphic to M%. (Henceforth we write ' = ' for isomorphic.) Thus we must ask a more restricted question. In particular we can ask to what extent is Mn determined by its homotopy type? We thus consider the following question. Let ^V{X) denote the equivalence classes of pairs (M, ƒ), where M is a compact smooth (or PL) «-manifold, ƒ: M -» X is a homotopy equivalence and (A/,/) ~ (M\ ƒ') if and only if there is an isomorphism M-* Af' making Received by the editors March 1, 1978. AMS (MOS) subject classifications (1970). Primary 14J10, 14J25, 32J15, 55A10, 57A15, 57A50, 57C25, 57C35, 57C45, 57D05, 57D10, 57D15, 57D60, 57D65, 57D80, 57E25; Secondary 55A25, 55A35, 57A10, 57D20, 57D40, 57D55. Key words and phrases. 4-manifold, 4-dimensional A-cobordism conjecture, complex surface, almost complete decomposability, framed link, calculus of links, Rohlin's theorem, ^-invariant, surgery, Whitney lemma, flexible handle, torus action, nonlocally flat PL-embedding, hand* lebody-decomposition. author partially supported by NSF grant MCS 77-04165. © 1980 American Mathematical Society 0002-9904/80/0000-0001/$40.75 1 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2 RICHARD MANDELBAUM the following diagram commute up to homotopy. ?SW(X) is thus the set of «-manifolds in the homotopy type of X'. We assume ^V{X) is nonempty and ask what can be said about ^"(X)*! If n = 1 or n = 2, then ^\Ln{X) will consist of only one element, namely the abstract surface with g-handles or with A>crosscaps if n = 2 and the circle S1 if n = 1. In fact if n = 2, G^Ct{X) is completely determined by its fundamen­ tal group irx (X), or even more computably by its first homology group Hx(X)l If n > 3 our question becomes much more difficult. To even begin to discuss it we focus our attention on the simpler situation which results if TTX(X) - 0. Thus suppose irx{X) = 0. Then if n = 3 our question is the classical Poincaré conjecture! Skipping n = 4 let us check what is known if n > 5. In this case our problem is the basic simply-connected surgery problem as attacked in [Br 1], [N]. Using surgery theory one can analyze the obstructions to deforming homotopy equivalences to isomorphisms. One obtains that for a given cohomology ring and choice of (rational) Pontryagin classes of a manifold M there exists only a finite number of manifolds ring-cohomology equivalent to M and having the same invariants. Furthermore fixing homo­ topy type and Pontryagin classes again gives only a finite number of isomor­ phism classes subordinate to the fixed homotopy type. Summarizing our discussion we arrive at the following diagram (where we include the equivalence relation of A-cobordism. Recall that Mn, Mm are cobordant if there exists a manifold Wn+l with dWn+l = M u M'. The triple (Wn+l, M, M') is called a cobordism. It is an A-cobordism if the inclusion M^> W is a homotopy equivalence and an ^-cobordism if i is a simple homotopy equivalence in the sense of Whitehead (see [M 5])). diffeomorphism 1 ^ f A-cobordism ] ^ f Homotopy type and j <*3 classes of mfds J I classes ƒ \ Pontryagin classes j Integral cohomology and 1 , . Pontryagin classes J where if n > 5 we have by the A-cobordism theorem that $t is 1-to-l, and by the preceding discussion that $2 and <J>3 are finite-to-one. What happens if n = 4? Then all the techniques used to establish the nature of $j, $2 and Oa above fail. As we shall see in Chapter 1, using entirely different methods one can show that for a simply-connected 4-mani- fold both 02 and 4>3 are in fact 1-to-l! However nothing is known about Q>x and it may even be infinite-to-one! It will shortly become apparent that four-dimensional manifolds occupy a peculiar position in the hierarchy of manifolds. On the one hand they are not License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use FOUR-DIMENSIONAL TOPOLOGY: AN INTRODUCTION 3 'sufficiently large' to allow one to use the higher-dimensional transversality and general position arguments which have proven so spectacularly successful in high-dimensional topology. Yet on the other hand their dimension is large enough to prevent a straightforward application of more intuitive lower- dimensional methods. This, coupled with the great gaps in our knowledge of three-dimensional topology, makes the subject of 4-manifolds a particularly difficult, but interesting one. This paper attempts to separate the two approaches. The first chapter establishes some basic theorems about 4-manifolds, while Chapters 2, 3, and parts of 7 explore an approach flavored by lower-dimensional intuition and analogy with lower-dimensional methods. The remaining chapters discuss modifications of higher-dimensional methods rendering them applicable to dimension 4. The paper seeks to be accurate in the statement of theorems, but is not rigorous in their proofs. Since the article originated as a survey talk, proofs of many theorems have been omitted while only geometric ideas and constructions have been sketched for others. Only a basic knowledge of modern topology is assumed, hence the somewhat long introductions and explanations which may be common knowledge to many readers. This paper could not have been written without the help of many people who influenced me in the course of my study of 4-manifolds. First I owe a great debt of thanks to Boris Moishezon who induced me to study 4-mani­ folds and with whom I spent many pleasant days studying their properties. I also would like to thank John Morgan, Robert Connelly and Laurent Sieben- mann for introducing me to the methods of PL-topology and showing me how topologists think about geometrical problems, as well as Bob Edwards, Cameron Gordon, Jose Montesinos, John Hollingsworth and Andrew Casson for many highly informative (as well as entertaining) conversations explaining various aspects of surgery and knot theory. In addition, thanks are due to Rob Kirby and his students Selman Akbulut, John Harer, Paul Melvin and Steve Kaplan for their invaluable aid in helping me understand the relation­ ship between link theory and 4-manifolds and to Sylvain Cappell and Julius Shaneson for spending much time explaining their own work on 4-manifolds and its relation to higher-dimensional phenomena. In addition Julius Shane- son read the original draft of this manuscript and helped me significantly in producing a hopefully more intelligible final version. Without the help of these mathematicians this paper would never have been written. Of course, I alone am responsible for any inaccuracies in the exposition. Clearly it is impossible to include all that is known about 4-manifolds in a survey article. I have essentially been guided by my own taste and thus tried to concentrate on areas which were unique from a four-dimensional point of view. I have, however excluded for the most part, results on wildness and taming, such as in [Mt 3], and those on Group actions on 4-manifold such as in ([Or 1], [Or 2], [Pao 1], [Pao 2], [Fin 1], [Fin 2], [FP]). Also I have mentioned only a few results on knottings of S2 in S4 which has a literature of its own. (See [Lorn] or [Fox].) Notation. By way of notation we will suppose that the basic notions of algebraic topology and differentiable manifolds, as given for example in [Lg] [Spr 1], are well known to the reader. We will also assume a knowledge of at License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 4 RICHARD MANDELBAUM least the elementary properties of piecewise-linear maps and manifolds and topological manifolds.
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