Introduction This book represents an attempt to collect and systematise the methods and main applications of the method of surgery, insofar as compact (but not nec- essarily connected, simply connected or closed) manifolds are involved. I have attempted to give a reasonably thorough account of the theoretical part, but have confined my discussion of applications mostly to those not accessible by surgery on simply connected manifolds (which case is easier, and already ade- quately covered in the literature). The plan of the book is as follows. Part 0 contains some necessary material (mostly from homotopy theory) and §1, intended as a general introduction to the technique of surgery. Part 1 consists of the statement and proof of our main result, namely that the possibility of successfully doing surgery depends on an obstruction in a certain abelian group, and that these 'surgery obstruction groups' depend only on the fundamental groups involved and on dimension modulo 4. Part 2 shows how to apply the result. §10 gives a rather detailed survey of the problem of classifying manifolds with a given simple homotopy type. In §11, we consider the analogous problem for submanifolds: it turns out that in codimension ^ 3 there are no surgery obstructions and in codimensions 1 and 2 the obstructions can be described by the preceding theory. Where alternative methods of studying these obstructions exist, we obtain calculations of surgery obstruction groups two such are obtained in §12. In part 3, I begin by summarising all methods of calculating surgery obstructions, and then apply some of these results to homeomorphism classification problems: my results on homotopy tori were used by Kirby and Siebenmann in their spectacular work on topological manifolds. In Part 4 are collected mentions of several ideas, half- formed during the writing of the book, but which the author does not have time to develop, and discussions of some of the papers on the subject which have been written by other authors during the last two years. The order of the chapters is not artificial, but readers who want to reach the main theorem as quickly as possible may find the following suggestions useful. Begin with §1, and read §4 next. Then glance at the statements in §3 and skip to §9 for the main part of the proof. Then read §10 and the first half of §11. Beyond this, it depends what you want: for the work on tori (§15), for example, you first need §12B, (13A.8) and (13B.8). The technique of surgery was not invented by the author, and this book clearly owes much to previous work by many others, particularly Milnor, Novikov and Browder. I have tried to give references in the body of the book wherever a result or proof is substantially due to someone else. xv
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