Transversality theorems for the weak topology

Let M and N be smooth manifolds. Denote by C ∞ (M, N ), the set of all smooth maps between M and N . The set C ∞ (M, N ) can then be given two topologies, the weak topology and the strong (Whitney) topology (see page 35 in Hirsch [6] for the definitions, see also [2] for more about the function space topologies). Many authors prefer to use a definition of this topology on the function spaces via jets; see page 42 in Golubitsky and Guillemin [4] and also a somewhat detailed discussion about the different topologies on function spaces in du Plessis and Vosegaard [1]. We will follow the approach of Hirsch [6]. Denote by C ∞ W (M, N ) and C ∞ S (M, N ), the space of smooth maps between M and N with the weak topology and the strong topology respectively. We prove the results for the weak topology and state them for the strong topology.

We say that a smooth map f : M → N is transverse to a submanifold S ⊂ N at x ∈ M , denoted f x S, if either f (x) / ∈ S or f (x) ∈ S and T f (x) S + Df x (T x M ) = T f (x) N . A map f is transverse to a submanifold on a subset K of the domain, denoted by f K Σ, if it is transverse at all points of the subset K. Notice that if the codimension of S is greater than the dimension of M then a map f : M → N is transverse to S if and only if f (M ) ∩ S = ∅, i.e., if the image of M under f is disjoint from S.

A major result that describes the transversal intersection property of smooth maps with respect to the submanifolds of the target manifold is the Thom transversality theorem (theorem 2.1 on page 74 in [6]). More precisely, Theorem 1.1. Let M and N be smooth manifolds, S ⊂ N a submanifold. Then, (a) T S = {f ∈ C ∞ (M, N ) : f S} is a dense subset of C ∞ W (M, N ) as well as of C ∞ S (M, N ).

(b) Suppose S is closed in N and K ⊂ M . Then {f ∈ C ∞ (M, N ) :

What would seem to be an obvious generalization of the transversality theorem 1.1 is to replace the submanifold S by a collection of submanifolds such that their union is a closed subset, which appears as exercise 3 on page 59 in Golubitsky and Guillemin’s book [4]. Unfortunately the exercise is not correct as stated and we will provide a counterexample later. A similar kind of mistake appears in exercise 8 on page 83 in Hirsch’s book [6]; we will also give a counterexample to this exercise.

In fact there seem to be no complete correct statements published to date of such results and perhaps due to the mistakes in the standard textbooks, many recent papers contain errors of a similar kind. See Trotman [10] for a brief discussion about some mistakes occurring in papers of Thom, Chenciner and Wall. More recently, Loi [8] proves a transversality theorem for definable maps in o-minimal structures, but his definition of the definable jet bundle is incorrect and due to the problems with definitions his proof is not correct as stated. Also, it is generally believed that the openness of the set of maps transverse to a stratification in the strong topology implies a-regularity of the stratification without any restrictions on the dimensions of manifolds, see for example remark 1.3.3 on page 38 in Goresky and Macpherson [5], but this is not true; see remark 1.5 below. In fact Goresky and Macpherson [5] also prove a more general transversality theorem, proposition 1.3.2 on page 38 and claim that the openness of maps, restricted to a stratification in the source manifold, transverse to a stratification in the target manifold in the strong topology implies that both stratifications are a-regular and refer to Trotman [10], but Trotman does not prove such a statement in his paper.

Here we prove a proposition which is more general than the transversality theorem and also gives the correct formulation of the incorrect exercises mentioned above. We need the following definitions:

A stratification Σ of a subset V of a manifold M is a locally finite partition of V into submanifolds of M . The submanifolds in the partition are called strata. By a locally finite partition we mean that each point of V has a neighbourhood meeting only finitely many strata.

Let S 1 and S 2 be two strata of Σ, S 2 is said to be a-regular over S 1 at x ∈ S 1 ∩S 2 if for every sequence of points {y i } in S 2 converging to x such that lim i→∞ T yi S 2 exists, we have

A stratification is called a-regular if every pair of strata (S i , S j ) is a-regular at every point in the intersections S i ∩ S j and S j ∩ S i .

Notice that our definition of a stratification allows that none of the strata be a closed subset; see figure 1. Both S 1 and S 2 are not closed and yet their union is closed. We say that a map f is transverse to a stratification Σ at a point x if f is transverse to each stratum S i at x. Now we state the result: Proposition 1.2. Let M and N be smooth manifolds and let Σ be an a-regular stratification of a closed subset S ⊂ N . Then, (a)

The strata of a stratification need not be closed submanifolds and they can be of same dimension but the union of the stra

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