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Surveys on Surgery Theory Volume 2
By Sylvain Cappell, Andrew Ranicki, Jonathan Rosenberg
PRINCETON UNIVERSITY PRESSCopyright © 2001 Princeton University Press
All rights reserved.
Surgery theory today—what it is and where it's going
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 of the main theorems and small simplifications here and there are definitely possible.
On the other hand, when it comes to applications of surgery theory, there have been many important recent developments in different directions, and as far as I know this is the first attempt to compare and contrast many of them.
To keep this survey within manageable limits, it was necessary to leave out a tremendous amount of very important material. So I needed to come up with selection criteria for deciding what to cover. I eventually settled on the following:
1. My first objective was to get across the major ideas of surgery theory in a non-technical way, even if it meant skipping over many details and definitions, or even oversimplifying the statements of major theorems.
2. My second objective was to give the reader some idea of the many areas in which the theory can be applied.
3. Finally, in the case of subjects covered elsewhere (and more expertly) in these volumes, I included a pointer to the appropriate article(s) but did not attempt to go into details myself.
I therefore beg the indulgence of the experts for the fact that some topics are covered in reasonable detail and others are barely mentioned at all.
I also apologize for the fact that the bibliography is very incomplete, and that I did not attempt to discuss the history of the subject or to give proper credit for the development of many important ideas. To give a complete history and bibliography of surgery would have been a very complicated enterprise and would have required a paper at least three times as long as this one.
I would like to thank Sylvain Cappell, Karsten Grove, Andrew Ranicki, and Shmuel Weinberger for many helpful suggestions about what to include (or not to include) in this survey. But the shortcomings of the exposition should be blamed only on me.
1 What is surgery?
1.1 The basics
Surgery is a procedure for changing one manifold into another (of the same dimension n) by excising a copy of Sr × Dn-r for some r, and replacing it by Dr+1 × Sn-r-1, which has the same boundary, Sr × Dn-r-1. This seemingly innocuous operation has spawned a vast industry among topologists. Our aim in this paper is to outline some of the motivations and achievements of surgery theory, and to indicate some potential future developments.
The classification of surfaces is a standard topic in graduate courses, so let us begin there. A surface is a 2-dimensional manifold. The basic result is that compact connected oriented surfaces, without boundary, are classified up to homeomorphism by the genus g (or equivalently, by the Euler characteristic χ = 2-2g). Recall that a surface of genus g is obtained from the sphere S2 by attaching g handles. The effect of a· surgery on S0 × D2 is to attach a handle, and of a surgery on S1 × D1 is to remove a handle. (See the picture on the next page.) Thus, from the surgery theoretic point of view, the genus g is the minimal number of surgeries required either to obtain the surface from a sphere, or else, starting from the given surface, to remove all the handles and reduce to the sphere S2. There is a similar surgery interpretation of the classification in the nonorientable case, with S2 replaced by the projective plane RP2.
In dimension n = 2, one could also classify manifolds up to home-omorphism by their fundamental groups, with 2g the minimal number of generators (in the orient able case). But for every n ≥ 4, every finitely presented group arises as the fundamental group of a compact n-manifold. It is not possible to classify finitely presented groups. Indeed, the problem of determining whether a finite group presentation yields the trivial group or not, is known to be undecidable. Thus there is no hope of a complete classification of all n-manifolds for n ≥ 4. Nevertheless, in many cases it is possible to use surgery to classify the manifolds within a given homotopy type, or even with a fixed fundamental group (such as the trivial group).
Just as for surfaces, high-dimensional manifolds are built out of handles. (In the smooth category, this follows from Morse theory. In the topological category, this is a deep result of Kirby and Siebenmann.) Again, each handle attachment or detachment is the result of a surgery. That is why surgery plays such a major role in the classification of manifolds. But since the same manifold may have many quite different handle decompositions, one needs an effective calculus for keeping track of the effect of many surgeries. This is what usually goes under the name of surgery theory.
Surgery theory has had remarkable successes. Here are some of the highlights:
the discovery and classification of exotic spheres (see  and );
the characterization of the homotopy types of differentiable manifolds among spaces with Poincaré duality of dimension ≥ 5 (Browder and Novikov; see in particular for an elementary exposition);
Novikov's proof of the topological invariance of the rational Pontrjagin classes (see );
the classification of "fake tori" (by Hsiang-Shaneson and by Wall) and of "fake projective spaces" (by Wall, also earlier by Rothenberg [unpublished] in the complex case): manifolds homotopy-equivalent to tori and projective spaces;
the disproof by Siebenmann of the manifold Hauptvermutung, the [false] conjecture that homeomorphic piecewise linear manifolds are PL-homeomorphic;
Kirby's proof of the Annulus conjecture and the work of Kirby and Siebenmann characterizing which topological manifolds (of dimension > 4) admit a piecewise linear structure;
the characterization (work of Wall, Thomas, and Madsen) of those finite groups that can act freely on spheres (the "topological space form problem" — see Section 3.5 below);
the construction and partial classification (by Cappell, Shaneson, and others) of "nonlinear similarities" (see 3.4.5 below), that is, linear representations of finite groups which are topologically conjugate but not linearly equivalent;
Freedman's classification of all simply-connected topological 4-manifolds, up to homeomorphism. (This includes the 4-dimensional topological Poincaré conjecture, the fact that all 4-dimensional homotopy spheres are homeomorphic to S4, as a special case.) For a survey of surgery theory as it applies to 4-manifolds, see .
the proof of Farrell and Jones of topological rigidity of compact locally symmetric spaces of non-positive curvature.
Surgery theory today
The main drawback of surgery theory is that it is necessarily quite complicated. Fortunately, one does not need to know everything about it in order to use it for many applications.
1.3 Dimension restrictions
As we have defined it, surgery is applicable to manifolds of all dimensions, and works quite well in dimension 2. The surgery theory novice is therefore often puzzled by the restriction in many theorems to the case of dimension ≥ 5. In order to do surgery on a manifold, one needs an embedded product of a sphere (usually in a specific homology class) and a disk. By the TUbular Neighborhood Theorem, this is the same as finding an embedded sphere with a trivial normal bundle. The main tool for constructing such spheres is the [strong] Whitney embedding theorem, which unfortunately fails for embeddings of surfaces into [smooth] 4-manifolds. This is the main source of the dimensional restrictions. Thus Smale was able to prove the h-cobordism theorem in dimensions ≥ 5, a recognition principle for manifolds, as well as the high-dimensional Poincaré conjecture, by repeated use of Whitney's theorem (and its proof). (See  for a nice exposition.) The h-cobordism theorem was later generalized by Barden, Mazur, and Stallings to the s-cobordism theorem for non-simply connected manifolds. This is the main tool, crucial for future developments, for recognizing when two seemingly different homotopy-equivalent manifolds are isomorphic (in the appropriate category, TOP, PL, or DIFF). The s-cobordism theorem is known to fail for 3-manifolds (where the cobordisms involved are 4-dimensional), at least in the category TOP, and for 4-manifolds, at least in the category DIFF (by Donaldson or Seiberg-Witten theory). Nevertheless, Freedman was able to obtain remarkable results on the topological classification of 4-manifolds by proving a version of Whitney's embedding theorem in the 4-dimensional topological category, with some restrictions on the fundamental group. This in turn has led to an s-cobordism theorem for 4-manifolds in TOP, provided that the fundamental groups involved have subexponential growth.
2 Tools of surgery
2.1 Fundamental group
The first topic one usually learns in algebraic topology is the theory of the fundamental group and covering spaces. In surgery theory, this plays an even bigger role than in most other areas of topology. Proper understanding of manifolds requires taking the fundamental group into account everywhere. As we mentioned before, any finitely presented group is the fundamental group of a closed manifold, but many interesting results of surgery theory only apply to a limited class of fundamental groups.
2.2 Poincaré duality
Any attempt to understand the structure of manifolds must take into account the structure of their homology and cohomology. The main phenomenon here is Poincaré[-Lefschetz] duality. For a compact oriented manifold Mn, possibly with boundary, this asserts that the cap product with the fundamental class [M, [partial derivative]M] [member of] Hn (M, [partial derivative];Z) gives an isomorphism
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (eq. 2.1)
This algebraic statement has important geometric content — it tells homologically how submanifolds of M intersect.
For surgery theory, one needs the generalization of Poincaré duality that takes the fundamental group π into account, using homology and cohomology with coefficients in the group ring Zπ. Or for work with non-orient able manifolds, one needs a still further generalization involving a twist by an orientation character w: π -> Z/2. The general form is similar to that in equation (eq. 2.1): one has a fundamental class [M, [partial derivative] M] [member of] Hn (M, [partial derivative] M; Z, w) and an isomorphism
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (eq. 2.2)
2.3 Hands-on geometry
One of Wall's great achievements (Chapter 5), which makes a general theory of non-simply connected surgery possible, is a characterization of when homology classes up to the middle dimension, in a manifold of dimension ≥ 5, can be represented by spheres with trivial normal bundles. This requires several ingredients. First is the Hurewicz theorem, which says that a homology class in the smallest degree where homology is nontrivial comes from the corresponding homotopy group, in other words, is represented by a map from a sphere. The next step is to check that this map is homotopic to an embedding, and this is where comes in. The third step requires keeping track of the normal bundle, and thus leads us to the next major tool:
2.4 Bundle theory
If X is a compact space such as a manifold, the m-dimensional real vector bundles over X are classified up to isomorphism by the homotopy classes of maps from X into BO(m), the limit (as k -> ∞) of the Grassmannian of m-dimensional subspaces of Rm+k. Identifying bundles which become isomorphic after the addition of trivial bundles gives the classification up to stable isomorphism, and amounts to replacing BO(m) by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. This has the advantage that [X, BO], the set of homotopy class; of maps X -> BO, is given by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], a cohomology theory. A basic fact is that if m exceeds the dimension of X, then one is already in the stable range, that is, the isomorphism classification of rank-m bundles over X coincides with the stable classification. Furthermore, if X is a manifold, then all embeddings of X into a Euclidean space of sufficiently high dimension are isotopic, by the [easy] Whitney embedding theorem, and so the normal bundle of X (for an arbitrary embedding into a Euclidean space or a sphere) is determined up to stable isomorphism. Thus it makes sense to talk about the stable normal bundle, which is stably an inverse to the tangent bundle (since the direct sum of the normal and tangent bundles is the restriction to X of the tangent bundle of Euclidean space, which is trivial).
Now consider a sphere Sr embedded in a manifold Mn. If 2r< n, then the normal bundle of Sr in Mn has dimension m = n - r > r and so is in the stable range, and hence is trivial if and only if it is stably trivial. Furthermore, since the tangent bundle of Sr is stably trivial, this happens exactly when the restriction to Sr of the stable normal bundle of Mn is trivial. If 2r = n, i.e., we are in the middle dimension, then things are more complicated. If M is oriented, then the Euler class of the normal bundle of Sr becomes relevant.
Poincaré duality, as discussed above in Section 2.2, naturally leads to the study of quadratic forms over the group ring Zπ of the fundamental group π. These are the basic building blocks for the definition of the surgery obstruction groups Ln (Zπ), which play a role in both the existence problem (when is a space homotopy-equivalent to a manifold?) and the classification problem (when are two manifolds isomorphic?). For calculational purposes, it is useful to define the L-groups more generally, for example, for arbitrary rings with involution, or for certain categories with an involution. The groups that appear in surgery theory are then important special cases, but are calculated by relating them to the groups for other situations (such as semisimple algebras with involution over a field). In fact the surgery obstruction groups for finite fundamental groups have been completely calculated this way, following a program initiated by Wall (e.g.,). For more details on the definition and calculation of the surgery obstruction groups by algebraic methods, see the surveys  and .
Algebra also enters into the theory in one more way, via Whitehead torsion (see the survey ) and algebraic K-theory. The key issue here is distinguishing between homotopy equivalence and simple homotopy equivalence, the kind of homotopy equivalence between complexes that can be built out of elementary contractions and expansions. These two notions coincide for simply connected spaces, but in general there is an obstruction to a homotopy equivalence being simple, called the Whitehead torsion, living in the Whitehead group Wh(π) of the fundamental group π of the spaces involved. This plays a basic role in manifold theory, because of the basic fact that if Mn is a manifold with dimension n ≥ 5 and fundamental group π, then any element of Wh(π) can be realized by an h-cobordism based on M, in other words, by a manifold Wn +1 with two boundary components, one of which is equal to M, such that the inclusion of either boundary component into W is a homotopy equivalence. In fact, this is just one part of the celebrated s-cobordism theorem, which also asserts that the h-cobordisms based on M, up to isomorphism (diffeomorphism if one is working with smooth manifolds, homeomorphism if one is working with smooth manifolds), are in bijection with Wh(π) via the Whitehead torsion of the inclusion [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The identity element of Wh(π) of course corresponds to the cylinder W = M × [0,1]. By the topological invariance of Whitehead torsion, any homeomorphism between manifolds is necessarily a simple homotopy equivalence, so Wh(π) is related to the complexity of the family of homeomorphism classes of manifolds homotopy equivalent to M. In addition, the Whitehead group is important for understanding "decorations" on the surgery obstruction groups, a technical issue we won't attempt to describe here at all.
Excerpted from Surveys on Surgery Theory Volume 2 by Sylvain Cappell, Andrew Ranicki, Jonathan Rosenberg. Copyright © 2001 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
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