Surveys on Surgery Theory: Volume 1. Papers Dedicated to C. T. C. Wall. (AM-145)

Sylvain Cappell
Andrew Ranicki
Jonathan Rosenberg
Pages: 448
https://www.jstor.org/stable/j.ctt7zv8q1

1. Front Matter
(pp. i-iv)
(pp. v-vi)
3. Preface
(pp. vii-2)
Sylvain Cappell, Andrew Ranicki and Jonathan Rosenberg
4. C. T. C. Wall’s contributions to the topology of manifolds
(pp. 3-16)
Sylvain Cappell, Andrew Ranicki and Jonathan Rosenberg

Numbered references in this survey refer to the Research Papers in Wall’s Publication List.

C. T. C. Wall¹ spent the first half of his career, roughly from 1959 to 1977, working in topology and related areas of algebra. In this period, he produced more than 90 research papers and two books, covering

cobordism groups,

the Steenrod algebra,

homological algebra,

manifolds of dimensions 3, 4, ≥ 5,

finiteness obstructions,

embeddings,

bundles,

Poincaré complexes,

surgery obstruction theory,

homology of groups,

2-dimensional complexes,

the topological space form problem,

computations ofK- andL-groups,

and more.

One quick measure of Wall’s influence...

5. C. T. C. Wall’s publication list
(pp. 17-24)

A geometric introduction to topology, vi, 168 pp., Addison Wesley, 1971; reprinted in paperback by Dover, 1993.

Surgery on compact manifolds, London Math. Soc. Monographs no. 1, x, 280 pp. Academic Press, 1970; 2nd edition (ed. A.A. Ranicki), Amer. Math. Soc. Surveys and Monographs 69, A.M.S., 1999.

Proceedings of Liverpool Singularities Symposium I., II.(edited), Lecture Notes in Math. 192, 209, Springer, 1971.

Homological group theory(edited), London Math. Soc. Lecture Notes 36, Cambridge University Press, 1979.

The geometry of topological stability(with A.A. du Plessis), viii, 572 pp., London Math. Soc. Monographs, New Series, no. 9, Oxford University Press,...

6. Classification of (n – 1)-connected 2n-dimensional manifolds and the discovery of exotic spheres
(pp. 25-30)
John Milnor

At Princeton in the fifties I was very much interested in the fundamental problem of understanding the topology of higher dimensional manifolds. In particular, I focussed on the class of 2n-dimensional manifolds which are (n– 1)-connected, since these seemed like the simplest examples for which one had a reasonable hope of progress. (Of course the class of manifolds with the homotopy type of a sphere is even simpler. However the generalized Poincaré problem of understanding such manifolds seemed much too difficult: I had no idea how to get started.) For a closed 2n-dimensional manifoldM2nwith no homotopy groups...

7. Surgery in the 1960’s
(pp. 31-40)
S. P. Novikov

I began to learn topology in 1956, mostly from Mikhail Mikhailovich Postnikov and Albert Solomonovich Schwartz (so by the end of 1957 I already knew much topology). I remember very well how Schwartz announced a lecture in 1957 in the celebrated “Topological Circle” of Paul Alexandrov entitled something like “On a differentiable manifold which does not admit a differentiable homeomorphism toS7”. Some topologists even regarded this title as announcing the discovery of non-differentiable manifolds homeomorphic toS7. In particular, this happened in the presence of my late father, who remarked that such a result contradicted his understanding of the...

8. Differential topology of higher dimensional manifolds
(pp. 41-72)
William Browder

These notes and the lectures they accompanied, were in no way intended to be any sort of treatise, or even rough outline of the subject of surgery theory at the time (1977). It was rather designed to give a little bit of the flavor of the subject to as wide an audience as possible, while at the same time giving an avenue, by way of the references, for someone to find their way into the subject in a serious way.

The standard model for a colloquium lecture (or lecture series) in mathematics is that the first quarter is understandable by...

9. Differentiable structures on manifolds
(pp. 73-104)
Timothy Lance

SupposeMis a closed smooth manifold. Is the “smoothness” of the underlying topological manifold unique up to diffeomorphism?

The answer is no, and the first, stunningly simple examples of distinct smooth structures were constructed for the 7-sphere by John Milnor as 3-sphere bundles overS4.

Theorem 1.1. (Milnor [52])For any odd integer k= 2j+ llet$M_{k}^{7}$be the smooth 7-manifold obtained by gluing two copies of D4×S3together via a map of the boundaries S3×S3given by${{f}_{j}}:(u,v)\to (u,{{u}^{1+j}}v{{u}^{-j}})$where the multiplication is quaternionic.Then$M_{k}^{7}$is homeomorphic to S7but, if k2≢ 1mod 7, is...

10. The Kervaire invariant and surgery theory
(pp. 105-120)
Edgar H. Brown Jr.

As an expository device we describe the development of this subject in chronological order beginning with Kervaire’s original paper ([10]) and Kervaire-Milnor’s Groups of Homotopy Spheres ([11]) followed by Frank Peterson’s and my work using Spin Cobordism ([5], [7]), Browder’s application of the Adams spectral sequence to the Kervaire invariant one problem ([3]), Browder-Novikov surgery ([16]) and finally an overall generalization of mine ([6]). In a final section we describe, with no detail, other work and references for these areas. We do not give any serious proofs until we get to the “overall generalization” sections where we prove the results...

11. A guide to the classification of manifolds
(pp. 121-134)
Matthias Kreck

The purpose of this note is to recall and to compare three methods for classifying manifolds of dimension ≥ 5.

The author has the impression that these methods are only known to a rather small group of insiders. This is related to the fact that the literature is not in good shape. If somebody gets interested in the classification problem, he has to go through a vast amount of literature until he perhaps finds out that the literature he has studied does not solve his specific problem. By presenting the basic principles of classification methods, the author hopes to provide...

12. Poincaré duality spaces
(pp. 135-166)
John R. Klein

At the end of the last century, Poincaré discovered that the Betti numbers of a closed oriented triangulated topologicaln-manifoldXn${{b}_{i}}(X):={{\dim}_{\mathbb{R}}}{{H}_{i}}(X;\mathbb{R})$satisfy the relation${{b}_{i}}(X)={{b}_{n-i}}(X)$(see e.g., [Di, pp. 21–22]). In modern language, we would say that there exists a chain map${{C}^{*}}(X)\to {{C}_{n-*}}(X)$which in every degree induces an isomorphism${{H}^{*}}(X)\cong {{H}_{n-*}}(X).$The original proof used the dual cell decomposition of the triangulation ofX. As algebraic topology developed in the course of the century, it became possible to extend the Poincaré duality theorem to non-triangulable topological manifolds, and also to homology manifolds.

In 1961, Browder [Br1] proved that...

13. Poincaré duality groups
(pp. 167-194)
Michael W. Davis

A spaceXisasphericalifπi(X) = 0 for alli> 1. For a space of the homotopy type of aCW-complex this is equivalent to the condition that its universal covering space is contractible.

Given any group Γ, there is an asphericalCW-complexBΓ (also denoted byK(Γ, 1)) with fundamental group Γ; moreover,BΓ is unique up to homotopy equivalence (cf. [Hu]).BΓ is called theclassifying spaceof Γ. (BΓ is also called anEilenberg-MacLane spacefor Γ.) So, the theory of asphericalCW-complexes, up to homotopy, is identical with the theory of groups. This point...

14. Manifold aspects of the Novikov Conjecture
(pp. 195-224)
James F. Davis

Let${{L}_{M}}\in {{H}^{4*}}(M;\mathbb{Q})$be the HirzebruchL-class of an oriented manifoldM. Let(orK(π, 1)) denote any aspherical space with fundamental groupπ. (A space is aspherical if it has a contractible universal cover.) In 1970 Novikov made the following conjecture.

Novikov Conjecture.Let$h:{{M}^{\prime }}\to M$be an orientation-preserving homotopy equivalence between closed, oriented manifoldsFor any discrete group π and any map f:M,${{f}_{*}}\circ {{h}_{*}}({{L}_{{{M}^{\prime }}}}\cap [{{M}^{\prime }}])={{f}_{*}}({{L}_{M}}\cap [M])\in {{H}_{*}}(B\pi ;\mathbb{Q})$

Many surveys have been written on the Novikov Conjecture. The goal here is to give an old-fashioned point of view, and emphasize connections with characteristic classes and the topology of manifolds. For...

15. A guide to the calculation of the surgery obstruction groups for finite groups
(pp. 225-274)
Ian Hambleton and Laurence R. Taylor

The surgery exact sequence of C. T. C. Wall [68] describes a method for classifying manifolds of dimension ≥ 5 within a given (simple) homotopy type, in terms of normal bundle information and a 4-periodic sequence of obstruction groups, depending only on the fundamental group and the orientation character. These obstruction groups$L_{n}^{s}(\text{Z}G,w)$are defined by considering stable isomorphism classes of quadratic forms on finitely generated free modules over ZG(neven), together with their unitary automorphisms (nodd).

Carrying out the surgery program in any particular case requires a calculation of the surgery obstruction groups, the normal invariants, and...

16. Surgery theory and infinite fundamental groups
(pp. 275-306)
C. W. Stark

We begin with a sketch of early work in surgery on manifolds with infinite fundamental groups, state some of the major problems in surgery and manifold topology concerning such manifolds, and then describe surgical aspects of certain classes and constructions of geometrically interesting infinite groups. The themes and outline of this survey may soon be outdated, especially as new aspects of the Borel and Novikov conjectures come to light.

Like many readers the author would be grateful for an authoritative history of these developments, but such an account is unlikely to appear since the intensity of activity around some of...

17. Continuously controlled surgery theory
(pp. 307-322)
Erik Kjær Pedersen

One of the basic questions in surgery theory is to determine whether a given homotopy equivalence of manifolds is homotopic to a homeomorphism. This can be determined by global algebraic topological invariants such as the normal invariant and the surgery obstruction (the Browder–Novikov–Sullivan–Wall theory). Another possibility is to impose extra geometric hypothesis on the homotopy equivalence. Such conditions are particularly useful when working in the topological category. Novikov’s proof of the topological invariance of the rational Pontrjagin classes only used that a homeomorphism is a homotopy equivalence with contractible point inverses, as was first observed by Sullivan....

18. Homology manifolds
(pp. 323-344)
Washington Mio

The study of the local-global geometric topology of homology manifolds has a long history. Homology manifolds were introduced in the 1930s in attempts to identify local homological properties that implied the duality theorems satisfied by manifolds [23, 56]. Bing’s work on decomposition space theory opened new perspectives. He constructed important examples of 3-dimensional homology manifolds with non-manifold points, which led to the study of other structural properties of these spaces, and also established hisshrinking criterionthat can be used to determine when homology manifolds obtained as decomposition spaces of manifolds are manifolds [4]. In the 1970s, the fundamental work...

19. A survey of applications of surgery to knot and link theory
(pp. 345-364)
Jerome Levine and Kent E. Orr

Knot and link theory studies how one manifold embeds in another. Given a manifold embedding, one can alter that embedding in a neighborhood of a point by removing this neighborhood and replacing it with an embedded disk pair. In this way traditional knot theory, the study of embeddings of spheres in spheres, impacts the general manifold embedding problem. In dimension one, the manifold embedding problemisknot and link theory.

This article attempts a rapid survey of the role of surgery in the development of knot and link theory. Surgery is one of the most powerful tools in dealing with...

20. Surgery and C*-algebras
(pp. 365-378)
John Roe

AC*-algebrais a complex¹ Banach algebraAwith an involution *, which satisfies the identity$\left\| {{x}^{*}}x \right\|={{\left\| x \right\|}^{2}}\forall x\in A.$

Key examples are

The algebraC(X) of continuous complex-valued functions on a compact Hausdorff spaceX.

The algebra$\mathfrak{B}(H)$of bounded linear operators on a Hilbert spaceH.

The theory ofC*-algebras began when Gelfand and Naimark proved that any commutativeC*-algebra with unit is of the formC(X), and a little while later that anyC*-algebra (commutative or not) is isomorphic to a subalgebra of some$\mathfrak{B}(H)$.

A simple consequence of Gelfand and Naimark’s characterisation of commutativeC*-algebras is the so-called...

21. The classification of Aloff-Wallach manifolds and their generalizations
(pp. 379-408)
R. James Milgram

The surgery program for classifying manifolds starts with the study of the homotopy type of the manifold and then applies the surgery exact sequence to determine thehorscobordism classes of manifolds within the homotopy type. There are different sequences depending on whether we work with piecewise linear or differential classification. In this note we apply the surgery program to study the classification of the set of free, isometricS1-actions on the Lie groupSU(3). Examples of these kinds originally occurred in surgery theory through the work of Kreck and Stoltz, [KSl], [KS2], motivated by results of Witten...

22. Elliptic cohomology
(pp. 409-439)
Charles B. Thomas

From the algebraic point of view elliptic cohomology is a quotient of spin cobordism, and as such forms part of a chain$\Omega _{spin}^{*}\to \cdots ?\to \varepsilon \ell \ell \to KSpi{{n}^{*}}\to {{H}^{*}},$with each link corresponding to a 1-dimensional commutative formal group law. In the case of elliptic cohomology this was written down by Euler in the 18th century, and the validity of the Eilenberg–Steenrod axioms follows from properties of addition on a class of elliptic curves in characteristicp.

In 1988, G. Segal gave a talk in the Bourbaki Seminar [Se], in which he summarised what was known at the time under the headings

(a)$\varepsilon \ell {{\ell }^{*}}(X)$is...