## Read an Excerpt

#### Surveys on Surgery Theory Volume 1

**By Sylvain Cappell, Andrew Ranicki, Jonathan Rosenberg**

**PRINCETON UNIVERSITY PRESS**

**Copyright © 2000 Princeton University Press**

All rights reserved.

ISBN: 978-0-691-04938-0

All rights reserved.

ISBN: 978-0-691-04938-0

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,

• 2-dimensional complexes,

• the topological space form problem,

• computations of *K-* and *L*-groups,

• 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 *n*-dimensional manifolds *Mn*1 and *Mn*2 are called *cobordant* if there is a compact manifold with boundary, say *Wn*+1 whose boundary is the disjoint union of *M*1 and *M*2. 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 *m*-dimensional manifold *M* and an embedding *Sr × Dm-r* [??] *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 × Dm-r* and gluing in *Sr × Dm-r* in its place:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The process of constructing *N* from *M* is called *surgery on an r-sphere,* 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 *h-cobordism* if the inclusions *M* [??] *W* and *N* [??] *W* are homotopy equivalences.

The importance of this notion stems from the *h-cobordism theorem* of Smale (ca. 1960), which showed that if *M* and *N* are simply connected and of dimension ≥ 5, then every *h*-cobordism 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 *h*-cobordant, and if *n* > 4, then *M* and *N* are diffeomorphic (or *PL*-homeomorphic, 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 *s-cobordism theorem,* which (under the same dimension restrictions) showed that the possible *h*-cobordisms between *M* and *N* are in natural bijection with the elements of the *Whitehead group* Wh π1 (*M*). The bijection sends an *h*-cobordism *W* to the *Whitehead torsion* of the associated homotopy equivalence from *M* to *W*, an invariant from algebraic *K*-theory that arises from the combinatorics of handle rearrangements. One consequence of this is that if *M* and *N* are *h*-cobordant 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 *h*-cobordism and s-cobordism theorems, became the foundation of Wall's work on manifold classification.

**4 4-Manifolds**

Milnor, following J. H. C. Whitehead, observed in 1956 that a simply connected 4-dimensional manifold *M* is classified up to homotopy equivalence by its *intersection form,* the non-degenerate symmetric bilinear from on *H2 (M;* Z) given by intersection of cycles, or in the dual picture, by the cup-product

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Note that the isomorphism *H*4 (*M*; Z) -> Z, and thus the form, depends on the orientation.

Classification of 4-dimensional 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 *h*-cobordism theorem for 4-dimensional manifolds:

**Theorem 4.1** (Wall [19]) *For two simply connected smooth closed oriented 4-manifolds M*1*and M*2, *the following are equivalent:*

*1. they are h-cobordant;*

*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 M*1 # *k*(*S*2 × *S*2) *and M*2 # *k*(*S*2 × *S*2) *are diffeomorphic (in a way preserving orientation) for k sufficiently large (depending on M*1*and M*2).

Note incidentally that the converse of the above theorem is not quite true: *M*1 # *k*(*S*2 × *S*2) and *M*2 # *k*(*S*2 × *S*2) are diffeomorphic (in a way preserving orientation) for *k* sufficiently large if and only if the intersection forms of *M*1 and *M*2 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 simply-connected 4-dimensional manifolds. Thanks to Donaldson's work, we now know that the stabilization in the theorem (with respect to addition of copies of *S*2 × *S*2) is unavoidable, in that without it, nothing like Theorem 4.1 could be true.

**5 Highly connected manifolds**

The investigation of simply connected 4-dimensional manifolds suggested the more general problem of classifying (*n* - 1)-connected 2*n*-dimensional 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 2*n*-dimensional manifold *M* with boundary a homology sphere [partial derivative]*M* = [summation]2*n*-1. 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* 2*n-dimensional manifolds with boundary a homotopy sphere are in natural bijection with the isomorphism classes of* Z-*valued non-degenerate* (-1)*n-symmetric forms with a quadratic refinement in* π*n(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 π*n*(*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 π*n(BSO(n)).* The quadratic refinement is defined by this homotopy class.)

The sequence of papers extended this diffeomorphism classification to other types of highly-connected 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 *Sq*-1 with *Y*/π finitely dominated, and that π acts freely on a *finite* complex homotopy equivalent to *Sq*-1 if and only if an algebraic *K*-theory 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 non-compact 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 non-simply-connected 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 *h*-cobordism (and hence, by Smale's theorem, up to diffeomorphism). If *Wn* is a parallelizable manifold with homotopy sphere boundary [partial derivative]*W* = [summation]*n*-1, 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 simply-connected finite *CW* complex *X* with *n*-dimensional Poincaré duality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is homotopy equivalent to a closed *n*-dimensional 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 degree-one 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] *P*4*k* = Z is one eighth of the difference between signature(*X*) and the 4*k*-dimensional component of the *L*-genus 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 homotopy-equivalent 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] *M*2*m*&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...)*

Excerpted fromSurveys on Surgery Theory Volume 1bySylvain Cappell, Andrew Ranicki, Jonathan Rosenberg. Copyright © 2000 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.

All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.

Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.