5 - Theory of handle decompositions

Summary

A handle decomposition is perhaps the simplest way to build a manifold from elementary pieces. The existence of such decompositions is obtained by analysing the geometry associated to a non-degenerate function on the manifold.

In the first section we prove the existence of handle decompositions for compact manifolds: in the next few sections we will show how to operate on such decompositions. In §5.2 we normalise the decomposition; then, after a section on the homology of handles, we manipulate the decompositions: there are results on adding handles, and on removing or introducing complementary pairs of handles. The technical details use the results treated in Chapter 2.

The definition of a handle decomposition is analogous to that of a CWcomplex. Also the results we establish run in parallel with operations on finite CW complexes that can be performed in homotopy theory. We will see below that up to a point the theory of handle presentations parallels that of cell decompositions and even to an important extent to that of algebraic operations on chain complexes.

The high point of this development is the h-cobordism theorem, which gives an effective criterion for diffeomorphism of compact manifolds. We prove this result in §5.5. Then we give a number of applications, discuss what is known in low dimensions, and outline what modifications need to be made to the theory when the fundamental group is non-trivial. In some places we anticipate Theorem 6.4.11, but Chapter 6 is independent of this chapter.

In this chapter, all manifolds will be compact unless otherwise stated.

Existence

Let W be a manifold, and suppose ∂₋W and ∂₊W disjoint manifolds with union ∂W. Then we call the pair (W, ∂₋W) a cobordism and the pair (W, ∂₊W) the dual cobordism; we also callW a cobordism of ∂₋W to ∂₊W, and say that ∂₋W and ∂₊W are cobordant. If W is a manifold with corner, and ∂₋W, ∂cW, ∂₊W are parts of the boundary such that ∂₋W and ∂₊W are disjoint and

we still call W a cobordism of ∂₋W to ∂₊W. We shall usually denote a cobordism by a single letter and often just call it a manifold. A picture of a cobordism is offered in Figure 5.1.