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Overview
Surgery theory, the basis for the classification theory of manifolds, is now about forty years old. There have been some extraordinary accomplishments in that time, which have led to enormously varied interactions with algebra, analysis, and geometry. Workers in many of these areas have often lamented the lack of a single source that surveys surgery theory and its applications. Indeed, no one person could write such a survey.
The sixtieth birthday of C. T. C. Wall, one of the leaders of the founding generation of surgery theory, provided an opportunity to rectify the situation and produce a comprehensive book on the subject. Experts have written stateoftheart reports that will be of broad interest to all those interested in topology, not only graduate students and mathematicians, but mathematical physicists as well.
Contributors include J. Milnor, S. Novikov, W. Browder, T. Lance, E. Brown, M. Kreck, J. Klein, M. Davis, J. Davis, I. Hambleton, L. Taylor, C. Stark, E. Pedersen, W. Mio, J. Levine, K. Orr, J. Roe, J. Milgram, and C. Thomas.
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Surveys on Surgery Theory Volume 1
By Sylvain Cappell, Andrew Ranicki, Jonathan Rosenberg
PRINCETON UNIVERSITY PRESS
Copyright © 2000 Princeton University PressAll rights reserved.
ISBN: 9780691049380
CHAPTER 1
C. T. C. Wall's contributions to the topology of manifolds
Sylvain Cappell, Andrew Ranicki, and Jonathan Rosenberg
Note. Numbered references in this survey refer to the Research Papers in Wall's Publication List.
1 A quick overview
C. T. C. Wall 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,
quadratic forms,
finiteness obstructions,
embeddings,
bundles,
Poincare complexes,
surgery obstruction theory,
homology of groups,
2dimensional complexes,
the topological space form problem,
computations of K and Lgroups,
and more.
One quick measure of Wall's influence is that there are two headings in the Mathematics Subject Classification that bear his name:
57Q12 (Wall finiteness obstruction for CW complexes).
57R67 (Surgery obstructions, Wall groups).
Above all, Wall was responsible for major advances in the topology of manifolds. Our aim in this survey is to give an overview of how his work has advanced our understanding of classification methods. Wall's approaches to manifold theory may conveniently be divided into three phases, according to the scheme:
1. All manifolds at once, up to cobordism (1959–1961).
2. One manifold at a time, up to diffeomorphism (1962–1966).
3. All manifolds within a homotopy type (1967–1977).
2 Cobordism
Two closed ndimensional manifolds Mn1 and Mn2 are called cobordant if there is a compact manifold with boundary, say Wn+1 whose boundary is the disjoint union of M1 and M2. Cobordism classes can be added via the disjoint union of manifolds, and multiplied via the Cartesian product of manifolds. Thorn (early 1950's) computed the cobordism ring N* of unoriented smooth manifolds, and began the calculation of the cobordism ring Ω* of oriented smooth manifolds.
After Milnor showed in the late 1950's that Ω* contains no odd torsion, Wall completed the calculation of Ω*. This was the ultimate achievement of the pioneering phase of cobordism theory. One version of Wall's main result is easy to state:
Theorem 2.1 (Wall [3]) All torsion in Ω*is of order 2. The oriented cobordism class of an oriented closed manifold is determined by its StiefelWhitney and Pontrjagin numbers.
For a fairly detailed discussion of Wall's method of proof and of its remarkable corollaries, see [Ros].
3 Structure of manifolds
What is the internal structure of a cobordism? Morse theory has as one of its main consequences (as pointed out by Milnor) that any cobordism between smooth manifolds can be built out of a sequence of handle attachments.
Definition 3.1 Given an mdimensional manifold M and an embedding Sr × Dmr [??] M, there is an associated elementary cobordism (W; M, N) obtained by attaching an (r + 1)handle to M × I. The cobordism W is the union
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],
and N is obtained from M by deleting Sr × Dmr and gluing in Sr × Dmr in its place:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
The process of constructing N from M is called surgery on an rsphere, or surgery in dimension r or in codimension m  r. Here r = 1 is allowed, and amounts to letting N be the disjoint union of M and Sm.
Any cobordism may be decomposed into such elementary cobordisms. In particular, any closed smooth manifold may be viewed as a cobordism between empty manifolds, and may thus be decomposed into handles.
Definition 3.2 A cobordism Wn+1 between manifolds Mn and Nn is called an hcobordism if the inclusions M [??] W and N [??] W are homotopy equivalences.
The importance of this notion stems from the hcobordism theorem of Smale (ca. 1960), which showed that if M and N are simply connected and of dimension ≥ 5, then every hcobordism between M and N is a cylinder M × I. The crux of the proof involves handle cancellations as well as Whitney's trick for removing double points of immersions in dimension > 4. In particular, if Mn and Nn are simply connected and hcobordant, and if n > 4, then M and N are diffeomorphic (or PLhomeomorphic, depending on whether one is working in the smooth or the PL category).
For manifolds which are not simply connected, the situation is more complicated and involves the fundamental group. But Smale's theorem was extended a few years later by Barden, Mazur, and Stallings to give the scobordism theorem, which (under the same dimension restrictions) showed that the possible hcobordisms between M and N are in natural bijection with the elements of the Whitehead group Wh π1 (M). The bijection sends an hcobordism W to the Whitehead torsion of the associated homotopy equivalence from M to W, an invariant from algebraic Ktheory that arises from the combinatorics of handle rearrangements. One consequence of this is that if M and N are hcobordant and the Whitehead torsion of the h cobordism vanishes (and in particular, if Wh π1(M) = 0, which is the case for many π1 's of practical interest), then M and N are again diffeomorphic (assuming n > 4).
The use of the Whitney trick and the analysis of handle rearrangements, crucial to the proof of the hcobordism and scobordism theorems, became the foundation of Wall's work on manifold classification.
4 4Manifolds
Milnor, following J. H. C. Whitehead, observed in 1956 that a simply connected 4dimensional manifold M is classified up to homotopy equivalence by its intersection form, the nondegenerate symmetric bilinear from on H2 (M; Z) given by intersection of cycles, or in the dual picture, by the cupproduct
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Note that the isomorphism H4 (M; Z) > Z, and thus the form, depends on the orientation.
Classification of 4dimensional manifolds up to homeomorphism or diffeomorphism, however, has remained to this day one of the hardest problems in topology, because of the failure of the Whitney trick in this dimension. Wall succeeded in 1964 to get around this difficulty at the expense of "stabilizing." He used handlebody theory to obtain a stabilized version of the hcobordism theorem for 4dimensional manifolds:
Theorem 4.1 (Wall [19]) For two simply connected smooth closed oriented 4manifolds M1and M2, the following are equivalent:
1. they are hcobordant;
2. they are homotopy equivalent (in a way preserving orientation);
3. they have the same intersection form on middle homology.
If these conditions hold, then M1 # k(S2 × S2) and M2 # k(S2 × S2) are diffeomorphic (in a way preserving orientation) for k sufficiently large (depending on M1and M2).
Note incidentally that the converse of the above theorem is not quite true: M1 # k(S2 × S2) and M2 # k(S2 × S2) are diffeomorphic (in a way preserving orientation) for k sufficiently large if and only if the intersection forms of M1 and M2 are stably isomorphic (where stability refers to addition of the hyperbolic form [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]).
From the 1960's until Donaldson's work in the 1980's, Theorem 4.1 was basically the only significant result on the diffeomorphism classification of simplyconnected 4dimensional manifolds. Thanks to Donaldson's work, we now know that the stabilization in the theorem (with respect to addition of copies of S2 × S2) is unavoidable, in that without it, nothing like Theorem 4.1 could be true.
5 Highly connected manifolds
The investigation of simply connected 4dimensional manifolds suggested the more general problem of classifying (n  1)connected 2ndimensional manifolds, for all n. The intersection form on middle homology again appears as a fundamental algebraic invariant of oriented homotopy type. In fact this invariant also makes sense for an (n  1)connected 2ndimensional manifold M with boundary a homology sphere [partial derivative]M = [summation]2n1. If [partial derivative]M is a homotopy sphere, it has a potentially exotic differentiable structure for n ≥ 4.
Theorem 5.1 (Wall) For n ≥ 3 the diffeomorphism classes of differentiable (n  1)connected 2ndimensional manifolds with boundary a homotopy sphere are in natural bijection with the isomorphism classes of Zvalued nondegenerate (1)nsymmetric forms with a quadratic refinement in πn(BSO(n)).
(The form associated to a manifold M is of course the intersection form on the middle homology Hn(M; Z). This group is isomorphic to πn(M), by the Hurewicz theorem, so every element is represented by a map Sn >M2n. By the Whitney trick, this can be deformed to an embedding, with normal bundle classified by an element of πn(BSO(n)). The quadratic refinement is defined by this homotopy class.)
The sequence of papers extended this diffeomorphism classification to other types of highlyconnected manifolds, using a combination of homotopy theory and the algebra of quadratic forms. These papers showed how far one could go in the classification of manifolds without surgery theory.
6 Finiteness obstruction
Recall that if X is a space and f: Sr >X is a map, the space obtained from X by attaching an (r + 1)cell is X [union]f Dr+1. A CW complex is a space obtained from 0 by attaching cells. It is called finite if only finitely many cells are used. One of the most natural questions in topology is:
When is a space homotopy equivalent to a finite CW complex?
A space X is called finitely dominated if it is a homotopy retract of a finite CW complex K, i.e., if there exist maps f: X >K, g: K >X and a homotopy gf [equivalent] 1 : X >X. This is clearly a necessary condition for X to be of the homotopy type of a finite CW complex. Furthermore, for spaces of geometric interest, finite domination is much easier to verify than finiteness. For example, already in 1932 Borsuk had proved that every compact ANR, such as a compact topological manifold, is finitely dominated. So another question arises:
Is a finitely dominated space homotopy equivalent to a finite CW complex?
This question also has roots in the study of the free actions of finite groups on spheres. A group with such an action necessarily has periodic cohomology. In the early 1960's Swan had proved that a finite group π with cohomology of period q acts freely on an infinite CW complex Y homotopy equivalent to Sq1 with Y/π finitely dominated, and that π acts freely on a finite complex homotopy equivalent to Sq1 if and only if an algebraic Ktheory invariant vanishes. Swan's theorem was in fact a special case of the following general result.
Theorem 6.1 (Wall) A finitely dominated space X has an associated obstruction [X] [member of] [??] (Z[π1(X)]). The space X is homotopy equivalent to a finite CW complex if and only if this obstruction vanishes.
The obstruction defined in this theorem, now universally called the Wall finiteness obstruction, is a fundamental algebraic invariant of noncompact topology. It arises as follows. If K is a finite CW complex dominating X, then the cellular chain complex of K, with local coefficients in the group ring Z[π1(X)], is a finite complex of finitely generated free modules. The domination of X by K thus determines a direct summand subcomplex of a finite chain complex, attached to X. Since a direct summand in a free module is projective, this chain complex attached to X consists of finitely generated projective modules. The Wall obstruction is a kind of "Euler characteristic" measuring whether or not this chain complex is chain equivalent to a finite complex of finitely generated free modules.
The Wall finiteness obstruction has turned out to have many applications to the topology of manifolds, most notably the Siebenmann end obstruction for closing tame ends of open manifolds.
7 Surgery theory and the Wall groups
The most significant of all of Wall's contributions to topology was undoubtedly his development of the general theory of nonsimplyconnected surgery. As defined above, surgery can be viewed as a means of creating new manifolds out of old ones. One measure of Wall's great influence was that when other workers (too numerous to list here) made use of surgery, they almost invariably drew upon Wall's contributions.
As a methodology for classifying manifolds, surgery was first developed in the 1961 work of Kervaire and Milnor [KM] classifying homotopy spheres in dimensions n ≥ 6, up to hcobordism (and hence, by Smale's theorem, up to diffeomorphism). If Wn is a parallelizable manifold with homotopy sphere boundary [partial derivative]W = [summation]n1, then it is possible to kill the homotopy groups of W by surgeries if and only if an obstruction
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],
vanishes. Here
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
In 1962 Browder [Br] used the surgery method to prove that, for n ≥ 5, a simplyconnected finite CW complex X with ndimensional Poincaré duality
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
is homotopy equivalent to a closed ndimensional differentiable manifold if and only if there exists a vector bundle η with spherical Thorn class such that an associated invariant σ [member of] Pn (the simply connected surgery obstruction) is 0. The result was proved by applying Thorn transversality to η to obtain a suitable degreeone map M >X from a manifold, and then killing the kernel of the induced map on homology. For n = 4k the invariant σ [member of] P4k = Z is one eighth of the difference between signature(X) and the 4kdimensional component of the Lgenus of η. (The minus sign comes from the fact that the tangent and normal bundles are stably the negatives of one another.) In this case, the result is a converse of the Hirzebruch signature theorem. In other words, X is homotopyequivalent to a differentiable manifold if and only if the formula of the theorem holds with η playing the role of the stable normal bundle. The hardest step was to find enough embedded spheres with trivial normal bundle in the middle dimension, using the Whitney embedding theorem for embeddings Sm [subset] M2m&mdahs;this requires π1(M) = {1} and m ≥ 3. Also in 1962, Novikov initiated the use of surgery in the study of the uniqueness of differentiable manifold structures in the homotopy type of a manifold, in the simplyconnected case.
(Continues...)
Table of Contents
The Editors Preface vii
The Editors C. T. C. Wall's contributions to the topology of manifolds 3
C. T. C. Wall's publication list 17
J. Milnor Classification of (n  l)connected 2ndimensional manifolds and the discovery of exotic spheres 25
S. Novikov Surgery in the 1960's 31
W. Browder Differential topology of higher dimensional manifolds 41
T. Lance Differentiable structures on manifolds 73
E. Brown The Kervaire invariant and surgery theory 105
A Kreck A guide to the classification of manifolds 121
J. Klein Poincare duality spaces 135
A Davis Poincare duality groups 167
J. Davis Manifold aspects of the Novikov Conjecture 195
I. Hambleton and L. Taylor A guide to the calculation of the surgery obstruction groups for finite groups 225
C. Stark Surgery theory and infinite fundamental groups 275
E. Pedersen Continuously controlled surgery theory 307
W. Mio Homology manifolds 323
J. Levine and K. Orr A survey of applications of surgery to knot and link theory 345
J. Roe Surgery and C*algebras 365
R. J. Milgram The classification of AloffWallach manifolds and their generalizations 379
C. Thomas Elliptic cohomology 409