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.
Previous Page Next Page