Surveys on Surgery Theory

Surveys on Surgery Theory: Volume 2. Papers Dedicated to C.T.C. Wall. (AM-149)

Sylvain Cappell
Andrew Ranicki
Jonathan Rosenberg
Copyright Date: 2001
Pages: 380
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  • Book Info
    Surveys on Surgery Theory
    Book Description:

    Surgery theory, the basis for the classification theory of manifolds, is now about forty years old. The sixtieth birthday (on December 14, 1996) of C.T.C. Wall, a leading member of the subject's founding generation, led the editors of this volume to reflect on the extraordinary accomplishments of surgery theory as well as its current enormously varied interactions with algebra, analysis, and geometry.

    Workers in many of these areas have often lamented the lack of a single source surveying surgery theory and its applications. Because no one person could write such a survey, the editors asked a variety of experts to report on the areas of current interest. This is the second of two volumes resulting from that collective effort. It will be useful to topologists, to other interested researchers, and to advanced students. The topics covered include current applications of surgery, Wall's finiteness obstruction, algebraic surgery, automorphisms and embeddings of manifolds, surgery theoretic methods for the study of group actions and stratified spaces, metrics of positive scalar curvature, and surgery in dimension four.

    In addition to the editors, the contributors are S. Ferry, M. Weiss, B. Williams, T. Goodwillie, J. Klein, S. Weinberger, B. Hughes, S. Stolz, R. Kirby, L. Taylor, and F. Quinn.

    eISBN: 978-1-4008-6521-5
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. i-iv)
  2. Table of Contents
    (pp. v-vi)
  3. Preface
    (pp. vii-2)
    Sylvain Cappell, Andrew Ranicki and Jonathan Rosenberg
  4. Surgery theory today — what it is and where it’s going
    (pp. 3-48)
    Jonathan Rosenberg

    This paper is an attempt to describe for a general mathematical audience what surgery theory is all about, how it is being used today, and where it might be going in the future. I have not hesitated to express my personal opinions, especially in Sections 1.2 and 4, though I am well aware that many experts would have a somewhat different point of view. Why such a survey now? The main outlines of surgery theory on compact manifolds have been complete for quite some time now, and major changes to this framework seem unlikely, even though better proofs of some...

  5. Reflections on C. T. C. Wall’s work on cobordism
    (pp. 49-62)
    Jonathan Rosenberg

    This book is intended to fulfill two main functions—to celebrate C. T. C. Wall’s contributions to topology, and to discuss the state of surgery theory today. This article concentrates on Wall’s first contribution to topology, still of great significance: the completion of the calculation of the cobordism ring Ω defined by René Thom. This subject, while not directly a part of surgery theory, is still vital to it. Surgery theory classifies manifolds (smooth, PL, or topological) within a given homotopy type, starting from the observation that one can construct new manifolds from old ones by means of surgery. It...

  6. A survey of Wall’s finiteness obstruction
    (pp. 63-80)
    Steve Ferry and Andrew Ranicki

    Wall’s finiteness obstruction is an algebraicK-theory invariant which decides if a finitely dominated space is homotopy equivalent to a finiteCWcomplex. The invariant was originally formulated in the context of surgery onCWcomplexes, generalizing Swan’s application of algebraicK-theory to the study of free actions of finite groups on spheres. In the context of surgery on manifolds, the invariant first arose as the Siebenmann obstruction to closing a tame end of a non-compact manifold. The object of this survey is to describe the Wall finiteness obstruction and some of its many applications to the surgery classification of...

  7. An introduction to algebraic surgery
    (pp. 81-164)
    Andrew Ranicki

    Surgery theory investigates the homotopy types of manifolds, using a combination of algebra and topology. It is the aim of these notes to provide an introduction to the more algebraic aspects of the theory, without losing sight of the geometric motivation.

    A closedm-dimensional topological manifoldMhas Poincaré duality isomorphisms

    Hm—*(M) ≅H*(M).

    In order for a spaceXto be homotopy equivalent to anm-dimensional manifold it is thus necessary (but not in general sufficient) forXto be anm-dimensional Poincaré duality space, withHm—*(X) ≅H*(X). Thetopological structure setSTOP(X) is defined...

  8. Automorphisms of manifolds
    (pp. 165-220)
    Michael Weiss and Bruce Williams

    This survey is about homotopy types of spaces of automorphisms of topological and smooth manifolds. Most of the results available arerelative, i.e., they compare different types of automorphisms.

    In chapter 1, which motivates the later chapters, we introduce our favorite types of manifold automorphisms and make a comparison by (mostly elementary) geometric methods. Chapters 2, 3, and 4 describe algebraic models (involvingL–theory and/or algebraicK–theory) for certain spaces of “structures” associated with a manifoldM, that is, spaces of other manifolds sharing some geometric features withM. The algebraic models rely heavily on

    Wall’s work in...

  9. Spaces of smooth embeddings, disjunction and surgery
    (pp. 221-284)
    Thomas G. Goodwillie, John R. Klein and Michael S. Weiss

    This survey traces the development, over more than 50 years, of a theory of smooth embeddings resting today on two pillars: the methods of disjunction and surgery. More precisely, the theory is about homotopical and homological properties of spaces of smooth embeddings emb(Mm,Nn). It is more satisfactory whennm≥ 3, but has something to offer in the other cases, too.

    Chapter 1 is about embeddings in the metastable range,m< 2n/3 approximately, and the idea of producing an embeddingMNby starting with an immersion and removing self-intersections. This goes back to Whitney [Wh2],...

  10. Surgery theoretic methods in group actions
    (pp. 285-318)
    Sylvain Cappell and Shmuel Weinberger

    This paper is intended to give a brief introduction to the applications of the ideas of surgery in transformation group theory; it is not intended to be any kind of survey of the latter theory, whose study requires many additional insights and methods. However, despite this disclaimer, there have been a number of signal achievements of the surgery theoretic viewpoint, notably in the directions of producing examples and, on occasion, giving complete classifications of particular sorts of actions.

    We have divided this paper into three sections which deal with three different variants of classical surgery. The first deals with “CW...

  11. Surgery and stratified spaces
    (pp. 319-352)
    Bruce Hughes and Shmuel Weinberger

    The past couple of decades has seen significant progress in the theory of stratified spaces through the application of controlled methods as well as through the applications of intersection homology. In this paper we will give a cursory introduction to this material, hopefully whetting your appetite to peruse more thorough accounts.

    In more detail, the contents of this paper are as follows: the first section deals with some examples of stratified spaces and describes some of the different categories that have been considered by various authors. For the purposes of this paper, we will work in either the PL category...

  12. Metrics of positive scalar curvature and connections with surgery
    (pp. 353-386)
    Jonathan Rosenberg and Stephan Stolz

    This chapter discusses the connection between geometry of Riemannian metrics of positive scalar curvature and surgery theory. While this is quite a deep subject which has attracted quite a bit of recent attention, the most surprising aspect of this whole area remains the original discovery of Gromov-Lawson and of Schoen-Yau from about 20 years ago—namely, that thereisa connection between positive scalar curvature metrics and surgery. The Surgery Theorem of Gromov-Lawson and Schoen-Yau remains the most important result in this subject. We discuss it and its variants at length in Section 3. Then in Section 4, we discuss...

  13. A survey of 4-manifolds through the eyes of surgery
    (pp. 387-422)
    Robion C. Kirby and Laurence R. Taylor

    Surgery theory is a method for constructing manifolds satisfying a given collection of homotopy conditions. It is usually combined with thes–cobordism theorem which constructs homeomorphisms or diffeomorphisms between two similar looking manifolds. Building on work of Sullivan, Wall applied these two techniques to the problem of computing structure sets. While this is not the only use of surgery theory, it is the aspect on which we will concentrate in this survey. In dimension 4, there are two versions, one in which one builds topological manifolds and homeomorphisms and the second in which one builds smooth manifolds and diffeomorphisms....

  14. Problems in 4-dimensional topology
    (pp. 423-437)
    Frank Quinn

    The early 1980’s saw enormous progress in understanding 4-manifolds: the topological Poincaré and annulus conjectures were proved, many cases of surgery and thes-cobordism theorem were settled, and Donaldson’s work showed that smooth structures are stranger than anyone had imagined. Big gaps remained: topological surgery ands-cobordisms with arbitrary fundamental group, and general classification results for smooth structures. Since then the topological work has been refined and applied, but the big problems are still unsettled. Gauge theory has flowered, but has had more to say about geometric structures (esp. complex or symplectic) than basic smooth structures. So on the foundational...