We open our discussion of the deeper properties of smooth manifolds with Whitney's embedding theorem for two reasons. The first is historical: smooth manifolds were originally considered as submanifolds of Euclidean spaces, and this theorem reconciled this approach with the abstract form of definition which we prefer. Secondly, the proof is quite simple, and opens the way to our later discussion of the general transversality theorem.

In Chapter 5 we will give a method for describing compact manifolds up to diffeomorphism. The method consists in defining a smooth function and then we can regard M as ‘filtered’ by the subset as a increases. In order to carry out this process in detail, it is necessary to suppose f non-degenerate. Thus we next give a direct proof of the existence of non-degenerate functions.

We proceed to techniques for moving a smooth map into ‘general position’. The language of jet spaces,which is basic to the study of singularities of smooth maps, is introduced in §4.4. Jets are also used to define topologies on function space (we give some proofs of properties of these topologies in §A.4).

The fundamental technical general position result is the transversality theorem, which is stated and proved in §4.5, and extended in the following section to multitransversality, to deal with the interaction of two maps with a common target. The development of transversality as a tool is due to Thom [150]; the very flexible formulation of multitransversality is due to Mather [88].

The main theorems include ‘general position’ results which we will often use in later chapters. In particular, a map may be supposed an embedding if (or an immersion if it may be deformed to avoid any subset of M of dimension, and to be transverse to any given submanifold of M.

However the results allow a much wider range of application: for example, dealing with transversality to submanifolds of jet space rather than just of M; and establishing that the set of smooth maps satisfying such conditions is open and dense in function space. We thus spend some time in §4.7 applying the main results to describe the singularities of a dense open set of maps when the target dimension is either small (≤ 2) or large (≥ 32 m).