In this chapter we discuss a method of constructing manifolds, or more precisely, of adapting a given manifold to satisfy certain conditions. This method is due to Milnor and Kervaire. In the paper [102] where they introduced the method, the objective was to simplify the homotopy type of themanifold, so the procedure was called ‘killing homotopy groups’. However since the procedure can be seen as removing a piece of a manifold and replacing it by something else, it has come to be known as ‘surgery’.

It was observed by Novikov that the method could be applied to the more general situation, given a manifold M and a map f : M → X, to change both M and f to make f more like a homotopy equivalence, by killing the homotopy groups of f. The method was then codified and further extended by Browder and by the author.

In more detail, the manifold M will be changed by a cobordism. As we saw in §5.1, we may choose a handle decomposition of this cobordism, so the procedure is broken into a sequence of operations, each corresponding to a single handle. Although we may think of M as a closed manifold, the discussion will apply to any compact manifold M.

In the first section we analyse a single step in the procedure: both the conditions for performing the step and its effect. In §7.2, we show how to modify a map f : M → X to kill all homotopy groups of f in dimensions below the middle.

In view of duality, any change to the homology of M is reflected by a corresponding change in the dual dimension. We next discuss the algebraic results we need on bilinear and quadratic forms, then in §7.4 formulate duality in the setting of CW-complexes.

In order to perform surgery to make f a homotopy equivalence, we must also require X to satisfy duality and it is convenient to suppose f a ‘normalmap’. As in Chapter 5, we discuss in detail in this book only the case when X is simplyconnected. We treat in turn the cases when the dimension of M is even (when there is an obstruction in Z or Z2 to performing surgery) or odd (when there is none).