Surgery theory today—what it is and where it's going

Jonathan Rosenberg

Introduction

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.

1.2 Successes

Surgery theory has had remarkable successes. Here are some of the highlights:

• the discovery and classification of exotic spheres (see [107] and [94]);

• 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 [110]);

• 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...