Proc. Conf. Opava (Czechoslovakia), August 24–28, 1992 Silesian University, Opava, 1993, 3–12

Openess and Density Theorems of Transversality in Manifolds with Corners ^{∗}

S. Armas-Gómez J. Margalef-Roig E. Outerelo-Domínguez E. Padrón-Fernández

Abstract

In this paper we deal with infinitesimal transversality, transversality, boundary-transver-sality and weak-boundary-transversality defined on manifolds with corners. We prove openess theorems about these types of transversality. The main theorem concerns the residuality of the jet-transversality (boundary-transversality and weak-boundary-transversality).

1991 Mathematics Subject Classification. Primary: 58C25, 58D15. Secondary: 58C27. Key words and phrases: Transversality, boundary-transversality, weak-boundary-transver-sality, infinitesimal transversality, Abraham-Quinn parametrized theorems of the density of the transversality, openess of transversality, Thom Density theorem of Transversality.

1 Introduction

We are interested in studying transversality of maps to a submanifold, especially openess and density of transversality. We deal with infinitesimal transversality [1], [2], transversality [8], boundary-transversality [3], and weak-boundary-transversality [4]. First we prove that

\{f ∈ C^{p}(X, X′) / for every x ∈ X, f(x) ∉ \overline{Y_{o}} or
f(x) ∈ \overline{Y_{o}} and f is infinitesimally transversal to Y at x\}

is an open set of C^{p}(X, X′) with the Whitney topology, where X and X′ are general manifolds (X, X′ can be infinite dimensional manifolds). From this result it follows that \{g ∈ C^{p}(X, X′) / for every x ∈ X_{i}^{h} with g(x) ∈ \overline{Y_{o}}, g_{|B_{h}X} ∩|_{x} Y \} is an open set of C^{p}(X, X′) with the Whitney topology, where X is finite dimensional, B_{h}X = ⋃_{i∈IN} X_{i}^{h} is a decomposition of B_{k}(X) by means of a stratification of compact submanifolds and \overline{Y_{o}} ⊆ Y. Thus if ∂X = ∅, we obtain that G is an open set of C^{p}(X, X′) with the Whitney topology, where

G = \{g ∈ C^{p}(X, X′) / for every x ∈ X, g(x) ∉ \overline{Y_{o}} or g(x) ∈ \overline{Y_{o}}, and g ∩| _{x}^{∗} Y \} =
= \{g ∈ C^{p}(X, X′) / for every x ∈ X, g(x) ∉ \overline{Y_{o}} or g(x) ∈ \overline{Y_{o}}, and g ∩| _{x}^{∂} Y \} =
= \{g ∈ C^{p}(X, X′) / for every x ∈ X, g(x) ∉ \overline{Y_{o}} or g(x) ∈ \overline{Y_{o}}, and g ∩|_{x} Y \}.

∗Supported by DGICYT-SPAIN, Proyecto PB89-0122 and Consejería de Educación del Gobierno de Canarias-SPAIN This paper is in final form and no version of it will be submitted for publication elsewhere.

Finally in the main theorems, 3.2 and 3.3, we prove the residuality of the sets {g ∈ C^{p}(X, X′) / j^{r}g ∩| Y }, {g ∈ C^{p}(X, X′) / j^{r}g ∩|^{∗} Y } and {g ∈ C^{p}(X, X′) / j^{r}g ∩|^{∂} Y }, with the Whitney topology, which generalize Thom’s theorem of the residuality of the jet-transversality [11].

2 Prerequisites

Throughout this paper manifolds will be Banach manifolds and they may have boundary if otherwise is not specified. Terminology and notations can be found in [8], [9] and [7].

Let X be a C^{p}-manifold and Y a C^{p}-submanifold, we say Y is a totally neat submanifold if ind_{Y} (x) = ind_{X}(x) for every x ∈ Y. If only ∂Y = Y ∩ ∂X, one says that Y is a neat submanifold.

Let f : X ⟶ X′ be a C^{p}-map and x_{0} ∈ X. The map f is called a submersion at x_{0} if there are an open neighbourhood V^{f(x_{0})} of f(x_{0}) in X′ and a C^{p}-map s : V^{f(x_{0})} ⟶ X such that s(f(x_{0})) = x_{0} and (f^{∘} s)(y) = y for all y ∈ V^{f(x_{0})}. We take this definition in order to have the following universal property: If f is a surjective submersion and g is a map, then g^{∘} f is a C^{p}-map if and only if g is a C^{p}-map. Nevertheless if f preserves the boundary at x, this definition is equivalent to the classical one: T_{x}f is surjective and its kernel admits a topological supplement in T_{x}X. Here, of course, f preserves the boundary at x means that there exists an open neighbourhood V^{x} of x in X such that f(V^{x} ∩ ∂X) ⊆ ∂X′. In any case the definition of submersion at x always implies the classical one.

The classical notion of transversality or infinitesimal transversality can be found in [1], [2], and the notion of transversality (local formulation) can be found in [8]. Next we explain the boundary and weak-boundary transversality ([3], [4]).

Definition 2.1 Let X, X′ be C^{p}-manifolds, f : X ⟶ X′ a C^{p}-map and Y′ a C^{p}-submanifold of X′. The map f is called boundary-transversal (respectively weakboundary-transversal) to Y′ at x ∈ X, and it will be denoted by f ∩| _{x}^{∂} Y′ (respectively f ∩| _{x}^{∗} Y′), if f(x) ∉ Y′ or f(x) ∈ Y′ and there are a chart c′ = (U′, φ′, (E′, Λ′)) of X′ adapted to Y′ at f(x) by means of (E′_{1}, Λ′_{1}), a topological supplement E′_{2} of E′_{1} in E′, a finite linearly independent system Λ′_{2} of L(E′_{2}, IR) and an open neighbourhood U of x in X, such that f(U) ⊆ U′,

(p′_{2} ^{∘} (θ′)^{−1} ^{∘} φ′ ^{∘} f)((U) ⊆ (E′_{2})_{Λ′_{2}}^{+}, p′_{2}(∂(E′_{1} × E′_{2})_{Λ′ ^{∘} θ′}^{+} ∩ ((θ′)^{−1} ^{∘} φ′ ^{∘} f)(∂U)) ⊆ ∂(E′_{2})_{Λ′_{2}}^{+}

and the C^{p}-map h : U ⟶ (E′_{2})_{Λ′_{2}}^{+}, defined by h(y) = (p′_{2} ^{∘} (θ′)^{−1} ^{∘} φ′ ^{∘} f_{|U})(y) for every y ∈ U is a submersion at x, (respectively f(U) ⊆ U′, (p′_{2} ^{∘} (θ′)^{−1} ^{∘} φ′ ^{∘} f)(U) ⊆ (E′_{2})_{Λ′_{2}}^{+} and the C^{p}-map h : U ⟶ (E′_{2})_{Λ′_{2}}^{+}, defined by h(y) = (p′_{2} ^{∘} (θ′)^{−1} ^{∘} φ′ ^{∘} f_{|U})(y) for every y ∈ U is a submersion at x) being θ′ : E′_{1} × E′_{2} ⟶ E′ the linear homeomorphism defined by θ′(a′, b′) = a′ +b′, p′_{2} the second projection from E′_{1} ×E′_{2} onto E′_{2} and Λ′ ^{∘} θ′ = {λ′ ^{∘} θ′ / λ′ ∈ Λ′}.

If A is a subset of X and f ∩| _{x}^{∂} Y′ (f ∩| _{x}^{∗} Y′) for every x ∈ A, we will say that f is boundary-transversal (weak-boundary-transversal) to Y′ along A and it will be denoted by f ∩| _{A}^{∂} Y′ (f ∩| _{A}^{∗} Y′). If A = X, one says f is boundary-transversal (weak-boundary-transversal) to Y′ and denotes by f ∩|^{∂} Y′ (f ∩|^{∗} Y′).

It is obvious that transversality [8] and boundary-transversality involve the weakboundary-transversality.

Proposition 2.1 [7] Let f : X ⟶ X′ be a C^{p}-map, Y′ a C^{p}-submanifold of X′ and x ∈ f^{−1}(Y′). Suppose that f ∩|_{x} Y′, (f ∩| _{x}^{∂} Y′ or f ∩| _{x}^{∗} Y′) then it holds:

(a) T_{f(x)}X′ = im(T_{x}f) + (T_{f(x)}j′)(T_{f(x)}Y′) and

(b) T_{x}f)^{−1}[(T_{f(x)}j′)(T_{f(x)}Y′)] admits a topological supplement in T_{x}X

where j′ : Y′ ↪ X′ is the inclusion map.

If (a) and (b) of the preceding proposition hold, we shall say that f is infinitesimally transversal to Y′ at x or briefly f is inf-t. to Y′ at x.

On the other hand, if x ∈ int(X), then f ∩|_{x} Y′ ⇔ f ∩| _{x}^{∗} Y′ ⇔ f ∩| _{x}^{∂} Y′. Moreover if x ∈ f^{−1}(Y′) ∩ int(X), one has f ∩| _{x}^{∗} Y′ if and only if f is infinitesimally transversal to Y′ at x.

If ind(x) = k, then f_{|B_{k}X} : B_{k}X ⟶ X′ is infinitesimally transversal to Y′ at x, (that is f_{|B_{k}X} ∩|_{x} Y′) if and only if f ∩|_{x} Y′ (see [8]).

Proposition 2.2 Let f : X → X′ be a C^{p}-map, Y a C^{p}-submanifold of X′ and x ∈ f^{−1}(Y). Then, if f preserves the boundary at x and (T_{f(x)}j′)(T_{f(x)}Y) ⊆ (T_{f(x)}X′)^{i}, the following statements are equivalent:

a) f is inf-t. to Y at x,  b) f ∩| _{x}^{∂} Y,  c) f ∩| _{x}^{∗} Y .

Let M, X and X′ be C^{p}-manifolds and ϱ a map from M into C^{p}(X, X′). One says ϱ is a C^{p}-representation if the map e_{ϱ} : M × X ⟶ X′, defined by e_{ϱ}(m, x) = ϱ(m)(x), is a C^{p}-map. In [4] we give a parametrized theorem of the density of the boundarytransversality.

Theorem 2.1 Let M, X and X′ be C^{p}-manifolds, p ≥ 1, ϱ : M → C^{p}(X, X′) a C^{p}- representation and W a C^{p}-submanifold of X′ which is a neat submanifold or is without boundary. Suppose that M is a T_{3} second countable space, X is a Hausdorff second countable space, dim_{x}(X) = n < +∞ for every x ∈ X, codim_{y}(W) = q < +∞ for every y ∈ W, p − 1 ≥ n − q, e_{ϱ} preserves the boundary at every z ∈ (e_{ϱ})^{−1}(W) and e_{ϱ} ∩|^{∂} W. Then {m ∈ M / ϱ(m) ∩|^{∂} W } is a residual set in M and therefore dense in M.

Let X, X′ be C^{p}-manifolds, p ≥ 1, k an integer, 0 ≤ k ≤ p, and J^{k}(X, X′) the set of k-jets from X to X′. Then there is a unique topology T^{(k)} on J^{k}(X, X′) such that for any two charts c = (U, φ, (E, Λ)) of X and c′ = (U′, φ′, (E′, Λ′)) of X′, the set J^{k}(U, U′) is open in T^{(k)} and the map:

π_{c}^{c′} : J^{k}(U, U′) ⟶ φ(U) × φ′(U′) × L(E, E′) ×… × L_{s}^{k}(E, E′)

defined by π_{c}^{c′}(j_{x}^{k}f) = (φ(x), φ′(f(x)), D(φ′ ^{∘} f^{∘} φ^{−1})(φ(x)),…, D^{k}(φ′ ^{∘} f^{∘} φ^{−1})(φ(x)) is a homeomorphism onto im(π_{c}^{c′}). Moreover if ∂X′ = ∅, there exists a unique differentiable structure [A] of class p − k on J^{k}(X, X′) such that for every chart c = (U, φ, (E, Λ)) of X and for every chart c′ = (U′, φ′, E′) of X′ the tuple d_{c}^{c′} = (J^{k}(U, U′), π_{c}^{c′}, (E × E′ × L(E, E′) ×… × L_{s}^{k}(E, E′), Λ^{∘} p_{1})) is a chart of (J^{k}(X, X′), [A]).

The topology of this manifold is the coherent topology T^{(k)}, the maps s : J^{k}(X, X′) → X and b : J^{k}(X, X′) → X′ defined by s(j_{x}^{k}f) = x and b(j_{x}^{k}f) = f(x) are surjective C^{p−k}-submersions and

∂^{q}J^{k}(X, X′) = s^{−1}(∂^{q}X), B_{q}J^{k}(X, X′) = s^{−1}(B_{q}X), q ≥ 0.

Definition 2.2 Let X, X′ be C^{p}-manifolds, p ∈ IN ∪ {∞}, k an integer with 0 ≤ k ≤ p and T_{s} the graph topology on C(X, J^{k}(X, X′)) (see [8]). Consider the map j^{k} : C^{p}(X, X′) → C(X, J^{k}(X, X′)) defined by (j^{k}(f))(x) = j_{x}^{k}f. Then the inverse image topology (j^{k})^{−1}(T_{s}), will be called Whitney topology of order k on C^{p}(X, X′) and will be denoted by T_{W} (p, k).

Then we have B_{W} (p, k) = {M(G) / G is an open set of J^{k}(X, X′) and M(G) ≠ ∅} is a basis of T_{W} (p, k), where M(G) = {f ∈ C^{p}(X, X′) / (j^{k}f)(X) ⊆ G}.

3 Openess and Density of Transversality

In [8] one makes a decomposition of a manifold without boundary by means of a stratification of compact submanifolds.

Proposition 3.1 Let X be a second countable Hausdorff non-compact C^{p}-manifold, p ∈ IN ∪ {∞}, with ∂X = ∅. Suppose that dim_{x}(X) = m ∈ IN for every x ∈ X. Then X = ⋃_{i∈IN} X_{i}, where X_{i} is a compact C^{p}-submanifold of X with ∂X_{i} ≠ ∅, ∂^{2}X_{i} = ∅, int(X_{i}) is an open set of X, X_{i} ⊆ Xi+1^{o} for every i ∈ IN. Moreover Xi^{o} = int(X_{i}) and dim(X_{i}) = m for every i ∈ IN, where Xi^{o} is the topological interior of X_{i} in X.

Corollary 3.1 Let X be a second countable Hausdorff C^{p}-manifold, p ∈ IN∪{∞}, with ∂X = ∅. Suppose that dim_{x}(X) = m ∈ IN ∪ {0} for every x ∈ X. Then X = ⋃_{i∈IN} X_{i}, where X_{i} is a compact C^{p}-submanifold of X with ∂^{2}X_{i} = ∅, X_{i} ⊆ X_{i+1}^{o}, int(X_{i}) is an open set of X and dim(X_{i}) = m for every i ∈ IN.

Proof: If X is compact, we take X_{i} = X for every i ∈ IN. If m ≥ 1 and X is not compact, it follows from the proposition 3.1. If m = 0 and X is not compact, then X is not finite, X = ⋃_{n∈IN}{x_{n}} and we take X_{n} = {x_{k} / 1 ≤ k ≤ n} for every n ∈ IN. □

Lemma 3.1 Let X, X′ be C^{p}-manifolds, p ∈ IN ∪ {∞}, f : X ⟶ X′ a C^{p}-map, Y′ a C^{p}-submanifold of X′, x ∈ f^{−1}(Y′), c = (U, φ, (E, Λ)) a chart of X centred at x, c′ = (U′, φ′, (E′, Λ′)) a chart of X′ adapted to Y′ at f(x) by means of (E′_{1}, Λ′_{1}) with f(U) ⊆ U′, E′_{2} a topological supplement of E′_{1} in E′ and h the map defined by:

h : φ(U) ⟶^{φ^{−1}} U ⟶^{f_{|U}} U′ ⟶^{φ′} φ′(U′) ⟶^{j′′} E′ ⟶^{(θ′)^{−1}} E′_{1} × E′_{2} ⟶^{p′_{2}} E′_{2}.

If y ∈ U ∩ f^{−1}(Y′), then we have that Dh(φ(y)) : E ⟶ E′_{2} is surjective and its kernel admits topological supplement in E if and only if f is inf-t. to Y at y.

Theorem 3.1 Let X, X′ be C^{p}-manifolds, p ∈ IN ∪ {∞}, Y a C^{p}-submanifold of X′, Y_{o} a subset of Y such that Y_{o} ≠ ∅, \overline{Y_{o}} ⊆ Y and r ∈ IN with 1 ≤ r ≤ p. Then

G=\{f ∈ C^{p}(X, X′) / for every x ∈ X, f(x) ∉ \overline{Y_{o}} or f(x) ∈ \overline{Y_{o}} and f is inf-t. to Y at x\}

is an open set of (C^{p}(X, X′), T_{W} (p, r)).

Proof: Let us consider

U^{r} = \{j_{x}^{r}f ∈ J^{r}(X, X′) / f(x) ∉ \overline{Y_{o}} or f(x) ∈ \overline{Y_{o}} and f is inf-t. to Y at x\}.

Let {j_{x_{n}}^{r}f_{n}}_{n∈IN} be a sequence in C = J^{r}(X, X′) − U^{r} which converges to j_{x_{o}}^{r}f_{o} ∈ J^{r}(X, X′) with the topology T^{(r)}. Consequently {x_{n}}_{n∈IN} converges to x_{o} and \{f_{n}(x_{n})\}_{n∈IN} converges to x′_{o} = f_{o}(x_{o}). For every n ∈ IN, f_{n}(x_{n}) = x′_{n} ∈ \overline{Y_{o}} and f_{n} is not infinitesimally transversal to Y at x_{n}. Then x′_{o} ∈ \overline{Y_{o}}.

Now we consider a chart c = (U, φ, (E, Λ)) of X centred at x_{o} with U ⊆ dom(f_{o}) and a chart c′ = (U′, φ′, (E′, Λ′)) of X′ adapted to Y at x′_{o} ∈ Y by means of (E′, Λ′) such that f_{o}(U) ⊆ U′. Let us consider the map h_{o} defined by

h_{o} : φ(U) ⟶^{φ^{−1}} U ⟶^{f_{o|U}} U′ ⟶^{φ′} φ′(U′) ⟶^{j′′} E′ ⟶^{θ^{−1}} E′_{1} × E′_{2} ⟶^{p_{2}} E′_{2},

where E′_{2} is a topological supplement of E′_{1} in E′. There exists n_{o} ∈ IN such that for every n ≥ n_{o}, there exists an open set V_{n} of φ(U) ⊆ E_{Λ}^{+} with φ(x_{n}) ∈ V_{n}, φ^{−1}(V_{n}) ⊆ dom(f_{n}) and f_{n}(φ^{−1}(V_{n})) ⊆ U′. Then we can define the map h_{n}, n ≥ n_{o}, as it follows:

h_{n} : V_{n} ⟶^{φ^{−1}} φ^{−1}(V_{n}) ⟶^{f_{n}} U′ ⟶^{φ′} φ′(U′) ⟶^{j′′} E′ ⟶^{θ^{−1}} E′_{1} × E′_{2} ⟶^{p_{2}} E′_{2},

We have {Dh_{n}(φ(x_{n}))}_{n∈IN} converges to Dh_{o}(φ(x_{o})) in L(E, E′_{2}). Suppose that f_{o} is infinitesimally transversal to Y at x_{o}. Then Dh_{o}(0) : E → E′_{2} is surjective and Ker(Dh_{o}(0)) admits a topological supplement in E. Therefore Dh_{o}(0) ∈ H = {l ∈ L(E, E′_{2}) / l is surjective and Ker(l) admits a topological supplement in E} which is open in L(E, E′_{2}). Thus there is m_{o} ≥ n_{o} such that Dh_{n}(φ(x_{n})) ∈ H for n ≥ m_{o}. By the lemma 3.1, for every n ≥ m_{o} ≥ n_{o}, f_{n} is infinitesimally transversal to Y at x_{n}, which is a contradiction. Then C is closed and M(U^{r}) is open in (C^{p}(X, X′), T_{W} (p, r)). Finally M(U^{r}) = G. □

Corollary 3.2 Let X, X′ be Hausdorff C^{p}-manifolds, p ∈ IN ∪ {∞}, where X is a second countable manifold. Suppose that dim_{x}(X) = m ∈ IN for every x ∈ X. Consider an integer r with 1 ≤ r ≤ p, a C^{p}-submanifold Y of X′, a subset Y_{o} of Y such that Y_{o} ≠ ∅, \overline{Y_{o}} ⊆ Y and B_{h}X = ⋃_{i∈IN} X_{i}^{h} for every h ∈ \{0,…, m\} as in the corollary 3.1 (notice that dim(B_{h}X) = m − h). Then for every i ∈ IN and every h ∈ {0,…, m},

G_{i}^{h} = \{g ∈ C^{p}(X, X′) / for every x ∈ X_{i}^{h} with g(x) ∈ \overline{Y_{o}}, g_{|B_{h}X} ∩|_{x} Y \}

is an open set of (C^{p}(X, X′), T_{W} (p, r)).

Proof: It is a consequence of the theorem 3.1, (note that T_{x}X_{i}^{h} = T_{x}B_{h}X) and of (since j : X_{i}^{h} ↪ X is a proper map) the continuity of the restriction map j^{∗} : C^{p}(X, X′) → C^{p}(X_{i}^{h}, X′) with the topology T_{W} (p, r). □

Corollary 3.3 In the hypotheses of the theorem 3.1, if we suppose that ∂X ∩f^{−1}(\overline{Y_{o}}) = ∅, then

G = \{f ∈ C^{p}(X, X′) / for every x ∈ X, f(x) ∉ \overline{Y_{o}} or f(x) ∈ \overline{Y_{o}} and f ∩|_{x} Y \} =
= \{f ∈ C^{p}(X, X′) / for every x ∈ X, f(x) ∉ \overline{Y_{o}} or f(x) ∈ \overline{Y_{o}} and f ∩| _{x}^{∗} Y \} =
= \{f ∈ C^{p}(X, X′) / for every x ∈ X, f(x) ∉ \overline{Y_{o}} or f(x) ∈ \overline{Y_{o}} and f ∩| _{x}^{∂} Y \},

is open of (C^{p}(X, X′), T_{W} (p, r)).

Proposition 3.2 In the conditions of the theorem 3.1, if G is the set defined in that theorem and we suppose that for every y ∈ \overline{Y_{o}}, (T_{y}j)(T_{y}Y) ⊆ (T_{y}X′)^{i}, where j : Y ↪ X′ is the inclusion map, then we have:

1. If C_{∂}^{p}(X, X′) = {f ∈ C^{p}(X, X′) / f preserves the boundary}, then G ∩ C_{∂}^{p}(X, X′) is an open set of (C_{∂}^{p}(X, X′), T_{W} (p, r)_{|C_{∂}^{p}(X,X′)}) and

G ∩ C_{∂}^{p}(X, X′) = \{f ∈ C_{∂}^{p}(X, X′) / for every x ∈ X, f(x) ∉ \overline{Y_{o}} or
f(x) ∈ \overline{Y_{o}} and f ∩| _{x}^{∂} Y \}.

2. If C_{∂,\overline{Y_{o}}}^{p}(X, X′) = \{f ∈ C^{p}(X, X′) / f preserves the boundary at every x ∈ f^{−1}(\overline{Y_{o}})\}, then G_{1} = G ∩ C_{∂,\overline{Y_{o}}}^{p}(X, X′) is an open set of (C_{∂,\overline{Y_{o}}}^{p}(X, X′), T_{W} (p, r)_{|C_{∂,\overline{Yo}}^{p}(X,X′)}) and

G_{1} = \{f ∈ C_{∂,\overline{Y_{o}}}^{p}(X, X′) / for every x ∈ X, f(x) ∉ \overline{Y_{o}} or f(x) ∈ \overline{Y_{o}} and f ∩| _{x}^{∂} Y \}
= \{f ∈ C_{∂,\overline{Y_{o}}}^{p}(X, X′) / for every x ∈ X, f(x) ∉ \overline{Y_{o}} or f(x) ∈ \overline{Y_{o}} and f ∩| _{x}^{∗} Y \}

Proposition 3.3 Let X, X′ be C^{p}-manifolds such that ∂X′ = ∅, p ∈ IN ∪ {∞} and r ∈ ZZ with 0 ≤ r < p. Let Y be a C^{p−r}-submanifold of J^{r}(X, X′), X_{o} a C^{p}-submanifold of X such that the inclusion j : X_{o} ↪ X is proper and t ∈ ZZ with 1 ≤ t ≤ p − r. Suppose that Y = ⋃_{m∈IN} Y_{m} with \overline{Y_{m}} ≠ ∅, \overline{Y_{m}} ⊆ Y for every m ∈ IN. Then:

1. If X and X′ are Hausdorff and X is paracompact,

(T_{m}^{X_{o}})^{r} = \{g ∈ C^{p}(X, X′) / for every x ∈ X_{o} with j_{x}^{r}g ∈ \overline{Y_{m}},
(j^{r}g)_{|X_{o}} is inf-t. to Y at x}

is an open set of (C^{p}(X, X′), T_{W} (p, t + r)), for every m ∈ IN.

2. T_{m}^{r} = \{g ∈ C^{p}(X, X′) / for every x ∈ X with j_{x}^{r}g ∈ \overline{Y_{m}}, j^{r}g is inf-t. to Y at x\} is an open set of (C^{p}(X, X′), T_{W} (p, t + r)), for every m ∈ IN.

3. {g ∈ C^{p}(X, X′) / j^{r}g is inf-t. to Y at x, for every x ∈ X with j_{x}^{r}g ∈ Y } = ⋂_{m∈IN} T_{m}^{r}.

Proof: 1. For every m ∈ IN we have, by the theorem 3.1,

H_{m} = \{h ∈ C^{p−r}(X_{o}, J^{r}(X, X′)) / for every x ∈ X_{o}, h(x) ∉ \overline{Y_{m}}
or h(x) ∈ \overline{Y_{m}} and h is inf-t. to Y at x\}

is an open set of (C^{p−r}(X_{o}, J^{r}(X, X′)), T_{W} (p − r, t)). Since the map j : X_{o} ↪ X is proper, the restriction map j^{∗} : C^{p−r}(X, J^{r}(X, X′)) ⟶ C^{p−r}(X_{o}, J^{r}(X, X′)) is continuous with T_{W} (p − r, t). So (j^{∗})^{−1}(H_{m}) is open in (C^{p−r}(X, J^{r}(X, X′)), T_{W} (p − r, t)). Finally (T_{m}^{X_{o}})^{r} = (j^{r})^{−1}(j^{∗})^{−1}(H_{m}) and j^{r} : (C^{p}(X, X′), T_{W} (p, t + r)) → (C^{p−r}(X, J^{r}(X, X′)), T_{W} (p − r, t)) defined by j^{r}(f)(x) = j_{x}^{r}f is continuous.

2 and 3 are easy to be checked. □

Corollary 3.4 Let X, X′ be a Hausdorff C^{p}-manifolds, p ∈ IN ∪ {∞} such that for every x ∈ X, dim_{x}(X) = n ∈ IN. Suppose that ∂X′ = ∅ and X is second countable. Let Y be a C^{p−r}-submanifold of J^{r}(X, X′) and t ∈ ZZ with 1 ≤ t ≤ p − r. If Y = ⋃_{m∈IN} Y_{m} with \overline{Y_{m}} ≠ ∅, \overline{Y_{m}} ⊆ Y for every m ∈ IN and B_{k}X = ⋃_{i∈IN} X_{i}^{k} for every k ∈ \{0,…, n\}, where X_{i}^{k} is a compact C^{p}-submanifold of B_{k}X with ∂^{2}X_{i}^{k} = ∅, int(X_{i}^{k}) is an open set of B_{k}X and dim(X_{i}^{k}) = n − k for every i ∈ IN, then for every m ∈ IN, i ∈ IN and k ∈ {0,…, n},

T (m, k, i) = \{g ∈ C^{p}(X, X′) / for every x ∈ X_{i}^{k} with j_{x}^{r}g ∈ \overline{Y_{m}}, (j^{r}g)_{|B_{k}X} ∩|_{x} Y \}

is an open set of (C^{p}(X, X′), T_{W} (p, t + r)) and {g ∈ C^{p}(X, X′) / j^{r}g ∩| Y } = ⋂_{m,k,i} T (m, k, i).

Proposition 3.4 With the same hypotheses and notation of the proposition 3.3, if for every y ∈ Y, (T_{y}j)(T_{y}Y) ⊆ (T_{y}J^{r}(X, X′))^{i} then

1. T_{m}^{r} = \{f ∈ C^{p}(X, X′) / for every x ∈ X, j_{x}^{r}f ∉ \overline{Y_{m}} or j_{x}^{r}f ∈ \overline{Y_{m}} and j^{r}f ∩| _{x}^{∂} Y \} =

= \{f ∈ C^{p}(X, X′ / for every x ∈ X, j_{x}^{r}f ∉ \overline{Y_{m}} or j_{x}^{r}f ∈ \overline{Y_{m}} and j^{r}f ∩| _{x}^{∗} Y \}.

2. ⋂_{m∈IN} T_{m}^{r} = {f ∈ C^{p}(X, X′) / j^{r}f ∩|^{∂} Y } = {f ∈ C^{p}(X, X′) / j^{r}f ∩|^{∗} Y }.

3. {g ∈ C^{p}(X, X′) / j^{r}g ∩| Y } ⊆ ⋂_{m∈IN} T_{m}^{r}.

Theorem 3.2 Let X, X′ be Hausdorff second countable C^{p}-manifolds, p ∈ IN ∪ {∞}, and r an integer, 0 ≤ r < p. Suppose that ∂X′ = ∅, dim_{x}(X) = n ∈ IN for every x ∈ X and dim_{x′}(X′) < +∞ for every x′ ∈ X′. Let Y be a C^{p−r}-submanifold of J^{r}(X, X′) without boundary and codim_{y}Y = q ∈ IN for every y ∈ Y, verifying p − r − 1 ≥ n − q and t ∈ ZZ with 1 ≤ t ≤ p − r. Then (T^{∂})_{Y}^{r} = {g ∈ C^{p}(X, X′) / j^{r}g ∩|^{∂} Y } is a residual, hence dense, set in (C^{p}(X, X′), T_{W} (p, r + t)). Moreover, (T^{∂})_{Y}^{r} = (T^{∗})_{Y}^{r} = {g ∈ C^{p}(X, X′) / j^{r}g ∩|^{∗} Y } ⊇ T_{Y}^{r} = {g ∈ C^{p}(X, X′) / j^{r}g ∩| Y }.

Proof: Since Y is a C^{p−r}-submanifold of J^{r}(X, X′), p − r > 0, then Y = G ∩ C where G is open and C is closed in J^{r}(X, X′) with the topology T^{(r)}. Moreover J^{r}(X, X′) is a Hausdorff, second countable, regular manifold.

For every y ∈ Y we take charts c_{y} = (U_{y}, φ_{y}, (IR^{n}, Λ_{y})), c′_{y} = (U′_{y}, φ′_{y}, IR^{n′_{y}}) of X and X′ centred at s(y) and b(y) respectively, such that \overline{U_{y}}, \overline{U′_{y}} are compact sets and φ_{y}(U_{y}), φ′_{y}(U′_{y}) are bounded sets. Let V_{y} be an open neighbourhood of y in J^{r}(X, X′) such that \overline{V_{y}} ⊆ G, \overline{V_{y}} is compact and (s, b)(\overline{V_{y}}) ⊆ U_{y} × U′_{y}. Then V_{y} ∩ Y = Y_{y} is an open set of Y, y ∈ Y_{y}, \overline{Y_{y}} ⊆ \overline{V_{y}} ∩ C ⊆ G ∩ C = Y, \overline{Y_{y}} is compact and (s, b)(\overline{Y_{y}}) ⊆ U_{y} × U′_{y}, where \overline{Y_{y}} is the adherence of Y_{y} in J^{r}(X, X′). Then Y = ⋃_{m∈IN} Y_{y_{m}} and, by the proposition 3.4 (notice that, for every y ∈ Y, T_{y}j(T_{y}Y) ⊆ (T_{y}J^{r}(X, X′))^{i} being j : Y ↪ X the inclusion map), for every m ∈ IN,

T_{m}^{r} = \{g ∈ C^{p}(X, X′) / for every x ∈ X with j_{x}^{r}g ∈ \overline{Y_{y_{m}}}, j^{r}g ∩| _{x}^{∂} Y \}

is an open set of (C^{p}(X, X′), T_{W} (p, t+r)) and {g ∈ C^{p}(X, X′) / j^{r}g ∩|^{∂} Y } = ⋂_{m∈IN} T_{m}^{r}.

We have (IR^{n})_{Λy_{m}}^{+} ⊇ φ_{y_{m}}(U_{y_{m}}) ⊇ φ_{y_{m}}(s(\overline{Y_{y_{m}}})) and therefore there exist open sets A_{m}, B_{m} of (IR^{n})_{Λy_{m}}^{+} such that φ_{y_{m}}(s(\overline{Y_{y_{m}}})) ⊆ A_{m} ⊆ \overline{A_{m}} ⊆ B_{m} ⊆ \overline{B_{m}} ⊆ φ_{y_{m}}(U_{y_{m}}) ⊆ (IR^{n})_{Λy_{m}}^{+} , (notice that (IR^{n})_{Λy_{m}}^{+} is a normal space) and a C^{∞}-map α_{m} : IR^{n} ⟶ [0, 1] such that α_{m}(\overline{A_{m}}) = \{1\} and α_{m}((IR^{n})_{Λy_{m}}^{+} − B_{m}) = \{0\}.

Analogously φ′_{y_{m}}(b(\overline{Y_{y_{m}}})) ⊆ φ′_{y_{m}}(U′_{y_{m}}) ⊆ IR^{n′_{ym}} and therefore there exist open sets A′_{m}, B′_{m} of IR^{n′_{ym}} such that φ′_{y_{m}}(b(\overline{Y_{y_{m}}})) ⊆ A′_{m} ⊆ \overline{A′_{m}} ⊆ B′_{m} ⊆ \overline{B′_{m}} ⊆ φ′_{y_{m}}(U′_{y_{m}}) ⊆ IR^{n′_{ym}} and a C^{∞}-map α′_{m} : IR^{n′_{ym}} ⟶ [0, 1] such that α′_{m}(y′) = 1 if y′ ∈ \overline{A′_{m}}, α′_{m}(y′) = 0 if y′ ∉ B′_{m}.

Consider f ∈ C^{p}(X, X′), M′_{m} = IR^{n′_{ym}} × L(IR^{n}, IR^{n′_{ym}}) ×… × L_{s}^{r}(IR^{n}, IR^{n′_{ym}}) and an open neighbourhood \overline{Y} _{y_{m}}^{∗} of \overline{Y_{y_{m}}} in Y such that φ_{y_{m}}(s(\overline{Y} _{y_{m}}^{∗})) ⊆ A_{m} and \overline{φ′_{y_{m}}(b(\overline{Y} _{y_{m}}^{∗})} ⊆ A′_{m}. Let us consider a compact set C_{m} of IR^{n} such that φ_{y_{m}}(U_{y_{m}}) ⊆ C_{m} ⊆ IR^{n} and the continuous map β_{m} defined by the following diagram:

β_{m} : IR^{n} × M′_{m} ⟶^{γ_{m}} IR^{n′_{ym}} ⟶^{∥ ∥} IR^{+} ∪ {0},

where γ_{m}(x, m′_{o}, m′_{1},…, m′_{r}) = m′_{o} + m′_{1}(x) + … + m′_{r}(x, …^{(r}, x).

Since \overline{B′_{m}} is a compact set, one has ε′_{m} = D(\overline{B′_{m}}, IR^{n′_{ym}} − φ′_{y_{m}}(U′_{y_{m}})) > 0 and there exists an open neighbourhood M_{m} of 0 in M′_{m} such that β_{m}(y, m′) < ε′_{m} for every y ∈ C_{m} and every m′ ∈ M_{m}. Hence if x ∈ U_{y_{m}} and f(x) ∈ U′_{y_{m}},

λ_{m}(m′, x) = α_{m}(φ_{y_{m}}(x)) · α′_{m}(φ′_{y_{m}}(f(x))) · m′(φ_{y_{m}}(x)) + φ′_{y_{m}}(f(x)) ∈ φ′_{y_{m}}(U′_{y_{m}})

for every m′ ∈ M_{m}, where m′ = (m′_{o}, m′_{1},…, m′_{r}) and m′(y) = ∑_{i=0}^{i=r} m′_{i}(y, …^{(i}, y).

Now we define the map ϱ_{m} : M_{m} ⟶ C^{p}(X, X′) by:

ϱ_{m}(m′)(x) = \{ \begin{matrix}f(x) & if & x ∉ U_{y_{m}} & or & f(x) ∉ U′_{y_{m}} \\ (φ′_{y_{m}})^{−1}(λ_{m}(m′, x)) & if & x ∈ U_{y_{m}} & and & f(x) ∈ U′_{y_{m}} \end{matrix}

Of course, since S_{m} = φ_{y_{m}}^{−1}(supp(α_{m}))∩f^{−1}((φ′_{y_{m}})^{−1}(supp(α′_{m}))) ⊆ U_{y_{m}} ∩f^{−1}(U′_{y_{m}}), S_{m} is closed in X and for every x ∈ U_{y_{m}} ∩ f^{−1}(U′_{y_{m}}) − S_{m}, (φ′_{y_{m}})^{−1}(λ_{m}(m′, x)) = f(x), we have that ϱ_{m}(m′) is a C^{p}-map. Moreover ϱ_{m} is a C^{p}-representation since e_{ϱ_{m}} : M_{m} × X ⟶ X′ is a C^{p}-map. Then the map j^{r}ϱ_{m} : M_{m} ⟶ C^{p−r}(X, J^{r}(X, X′)) defined by (j^{r}ϱ_{m})(m′) = j^{r}ϱ_{m}(m′), is a C^{p−r}-representation, which follows easily by taking charts.

Let us consider

ε_{m} = \frac{1}{2}min\{D(supp(α′_{m}), IR^{n′_{ym}} − φ′_{y_{m}}(U′_{y_{m}})), D(\overline{φ′_{y_{m}}(b(\overline{Y} _{y_{m}}^{∗}))}, IR^{n′_{ym}} − A′_{m})\} and M_{m}^{∗} = {m′ ∈ M_{m} / ∥m′(φ_{y_{m}}(x))∥ < ε_{m} for every x ∈ U_{y_{m}}such that φ_{y_{m}}(x) ∈ supp(α_{m})}.

Then M_{m}^{∗} is an open set of M_{m} and 0 ∈ M_{m}^{∗}. Let us consider (m′, x) ∈ (M_{m}^{∗} × X) ∩ (e_{j^{r}ϱ_{m}})^{−1}(\overline{Y} _{y_{m}}^{∗}). We have the pair (x, ϱ_{m}(m′)(x)) belongs to U_{y_{m}} × U′_{y_{m}} and φ_{y_{m}}(x) ∈ A_{m}.

On the other hand f(x) ∈ U′_{y_{m}} and we have

τ_{m} = d(φ′_{y_{m}}(f(x)), φ′_{y_{m}}(ϱ_{m}(m′)(x))) = d(φ′_{y_{m}}(f(x)), λ_{m}(m′, x)) ≤ ∥m′(φ_{y_{m}}(x))∥ < ε_{m}.

Since φ′_{y_{m}}(ϱ_{m}(m′)(x)) ∈ φ′_{y_{m}}(b(\overline{Y} _{y_{m}}^{∗})) and ε_{m} < D(\overline{φ′_{y_{m}}(b(\overline{Y} _{y_{m}}^{∗}))}, IR^{n′_{ym}} − A′_{m}), we have φ′_{y_{m}}(f(x)) ∈ A′_{m}. Since φ′_{y_{m}}(f(x)) ∈ A′_{m} and φ_{y_{m}}(x) ∈ A_{m}, there exists an open neighbourhood V_{m}^{x} of x in (φ_{y_{m}})^{−1}(A_{m}) ∩ f^{−1}(U′_{y_{m}}) such that φ′_{y_{m}}(f(V_{m}^{x})) ⊆ A′_{m}. Thus for every (m′′, y) ∈ M_{m}^{∗} × V_{m}^{x}, we have e_{j^{r}ϱ_{m}}(m′′, y) = j_{y}^{r}ϱ_{m}(m′′) ∈ J^{r}(V_{m}^{x}, U′_{y_{m}}) and

φ′_{y_{m}}(ϱ_{m}(m′′)(y)) = λ_{m}(m′′, y) = m′′(φ_{y_{m}}(y)) + φ′_{y_{m}}(f(y))

Consider the following commutative diagram:

[figure: M_{m}^{∗} × V_{m}^{x}; J^{r}(V_{m}^{x}, U′_{y_{m}}); e_{j^{r}ϱ_{m}}; M_{m}^{∗} × φ_{y_{m}}(V_{m}^{x}); φ_{y_{m}}(V_{m}^{x}) × φ′_{y_{m}}(U′_{y_{m}}) × L(IR^{n}, IR^{n′_{ym}}) ×… × L_{s}^{r}(IR^{n}, IR^{n′_{ym}}); 1 × φ_{y_{m}}; π_{c_{ym}}^{c′_{ym}}; h]

where h(m′′, y) = (y, t_{m′′}(y), Dt_{m′′}(y),…, D^{r}t_{m′′}(y)), being

t_{m′′}(y) = m′′(y) + φ′_{y_{m}}(f(φ_{y_{m}}^{−1}(y))) = φ′_{y_{m}}(ϱ_{m}(m′′)(φ_{y_{m}}^{−1}(y)))

for every m′′ ∈ M_{m}^{∗} and every y ∈ φ_{y_{m}}(V_{m}^{x}). Then the map defined similarly to h

\overline{h} : M′_{m} × φ_{y_{m}}(V_{m}^{x}) ⟶ φ_{y_{m}}(V_{m}^{x}) × IR^{n′_{ym}} × L(IR^{n}, IR^{n′_{ym}}) ×… × L_{s}^{r}(IR^{n}, IR^{n′_{ym}})

is a C^{p−r}-diffeomorphism. Indeed,

D^{i}t_{m′}(y) = ∑_{k=0}^{r−i} \frac{(i + k)!}{k!}m′_{i+k}(., ., .,^{(i} y, …^{(k} y) + D^{i}λ(y)

for every i = 1,…, r, being λ = φ′_{y_{m}} ^{∘} f^{∘} (φ_{y_{m}})^{−1} and m′ = (m′_{o},…, m′_{r}). It is clear that \overline{h} is bijective.

Moreover (\overline{h})^{−1}(\overline{y}, \overline{m_{o}}, \overline{m}_{1},…, \overline{m_{r}}) = (m′_{o}, m′_{1},…, m′_{r}, y), where y = ȳ and for every i = 0,…, r

m′_{i} = \frac{1}{i!} ∑_{p=0}^{r−i}(−1)^{p} \frac{1}{p!}[\overline{m}_{i+p}(., ., .,^{(i} ȳ, …^{(p} ȳ) − D^{i+p}λ(ȳ)(., ., .,^{(i} ȳ, …^{(p} ȳ)].

Thus h is a diffeomorphism from M_{m}^{∗} × φ_{y_{m}}(V_{m}^{x}) onto its image and therefore e_{j^{r}ϱ_{m}} : M_{m}^{∗} × V_{m}^{x} → J^{r}(V_{m}^{x}, U′_{y_{m}}) is a C^{p−r}-diffeomorphism onto its image which is an open set. Then e_{j^{r}ϱ_{m}} : M_{m}^{∗} × X → J^{r}(X, X′) is infinitesimally transversal to \overline{Y} _{y_{m}}^{∗} at every (m′, x) ∈ M_{m}^{∗} × X such that e_{j^{r}ϱ_{m}}(m′, x) = j_{x}^{r}(ϱ_{m}(m′)) ∈ \overline{Y} _{y_{m}}^{∗} and, by the proposition 2.2, e_{j^{r}ϱ_{m}} ∩|^{∂} \overline{Y} _{y_{m}}^{∗} and e_{j^{r}ϱ_{m}} ∩|^{∗} \overline{Y} _{y_{m}}^{∗}. Thus, by the theorem 2.1, K = \{m′ ∈ M_{m}^{∗} / j^{r}ϱ_{m}(m′) ∩|^{∂} \overline{Y} _{y_{m}}^{∗}\} is residual, hence dense, in M_{m}^{∗}. Then there exists a sequence {m′_{p}}_{p∈IN} of elements of K which converges to 0.

Now we have that ϱ_{m}(m′_{p})_{|X−\overline{U_{ym}}} = f_{X−\overline{U_{ym}}} for every p ∈ IN. On the other hand the map j^{r+t}ϱ_{m} : M_{m}^{∗} ⟶ C^{p−(r+t)}(X, J^{r+t}(X, X′)), defined by (j^{r+t}ϱ_{m})(m′) = j^{r+t}ϱ_{m}(m′), is continuous with the compact-open topology because e_{j^{r+t}ϱ_{m}} : M_{m}^{∗} × X ⟶ J^{r+t}(X, X′) is continuous . Thus {j^{r+t}ϱ_{m}(m′_{p})}_{p∈IN} converges to j^{r+t}ϱ_{m}(0) = j^{r+t}f with the compact-open topology and \{j^{r+t}ϱ_{m}(m′_{p})_{|\overline{U_{ym}}} \}_{p∈IN} converges uniformly to (j^{r+t}f)\overline{U_{ym}} .

Now {ϱ_{m}(m′_{p})}_{p∈IN} converges to ϱ_{m}(0) = f in (C^{p}(X, X′), T_{W} (p, r + t)) (see [8]). Consequently \{g ∈ C^{p}(X, X′) / j^{r}g ∩|^{∂} \overline{Y} _{y_{m}}^{∗}\} is a dense set in (C^{p}(X, X′), T_{W} (p,

r + t)). Finally \{g ∈ C^{p}(X, X′) / j^{r}g ∩|^{∂} \overline{Y} _{y_{m}}^{∗}\} ⊆ T_{m}^{r},and \{g ∈ C^{p}(X, X′) / j^{r}g ∩|^{∂} Y \} = {g ∈ C^{p}(X, X′) / j^{r}g ∩|^{∗} Y } = ⋂_{m∈IN} T_{m}^{r}. □

If we have “Y is a neat C^{p−r}-submanifold of J^{r}(X, X′) such that, for every y ∈ Y, (T_{y}j)(T_{y}Y) ⊆ (T_{y}J^{r}(X, X′))^{i}” then “Y is a C^{p−r}-submanifold of J^{r}(X, X′) without boundary”.

For transversality, using the parametrized density theorem [8] and the corollaries 3.1 and 3.4, in the same way that the above theorem, one obtains the following result [8].

Theorem 3.3 Let X, X′ be Hausdorff second countable C^{p}-manifolds, p ∈ IN ∪ {∞}, and r an integer, 0 ≤ r < p. Suppose that ∂X′ = ∅, dim_{x}(X) = n ∈ IN for every x ∈ X and dim_{x′}(X′) < +∞ for every x′ ∈ X′. Let Y be a totally neat C^{p−r}-submanifold of J^{r}(X, X′) with constant finite codimension q verifying p − r − 1 ≥ n − q and t ∈ ZZ with 1 ≤ t ≤ p − r. Then T_{Y}^{r} = {g ∈ C^{p}(X, X′) / j^{r}g ∩| Y } is a residual, hence dense, set in (C^{p}(X, X′), T_{W} (p, r + t)). Moreover, T_{Y}^{r} ⊆ (T^{∗})_{Y}^{r} = {g ∈ C^{p}(X, X′) / j^{r}g ∩|^{∗} Y }, and therefore (T^{∗})_{Y}^{r} is a residual, hence dense, set of (C^{p}(X, X′), T_{W} (p, r + t)).

The propositions 2.2, 3.2, 3.4 and the theorem 3.2 are also true if we put “there exist a topological supplement F_{y}^{∗} of (T_{y}j)(T_{y}Y) in T_{y}Z and a finite linearly independent system Λ_{y}^{∗} of L(F_{y}^{∗}, IR) such that (T_{y}Z)^{i} ⊆ (T_{y}j)(T_{y}Y) + (F_{y}^{∗})_{Λ^{∗}_{y}}^{+} and ∂(T_{y}Z)^{i} ⊆ (T_{y}j)(T_{y}Y) + ∂(F_{y}^{∗})_{Λ^{∗}_{y}}^{+},” instead of “(T_{y}j)(T_{y}Y) ⊆ (T_{y}Z)^{i}”, being Z = X′ in the propositions 2.2, 3.2 and Z = J^{r}(X, X′) in the proposition 3.4 and the theorem 3.2 (see [4]).

References

[1] Abraham R., Transversality on manifolds of mappings, Bull. A.M.S. 69 (1963), 470–474.

[2] Abraham-Robbin, “Transversal Mappings and Flows”, Benjamin, 1967.

[3] Armas-Gómez S., Margalef-Roig J., Outerelo-Domínguez E., Padrón-Fernández E., Transversality on manifolds with corners (Preprint).

[4] Armas-Gómez S., Margalef-Roig J., Outerelo-Domínguez E., Padrón-Fernández E., Transversality on infinite dimensional manifolds with corners (Preprint).

[5] Golubitsky M., Guillemin V., “Stable Mappings and their Singularities”, Springer-Verlag, GTM 14, 1973.

[6] Margalef-Roig J., Outerelo-Domínguez E., Density of the transversality on manifolds, Collectanea Mathematica 37 (1986), 277–285.

[7] Margalef-Roig J., Outerelo-Domínguez E., “Topologia Diferencial”, Consejo Superior de Investigaciones Cientificas, Madrid, 1988.

[8] Margalef-Roig J., Outerelo-Domínguez E., “Differential Topology”, North-Holland, Mathematics Studies 173, Amsterdam, 1992.

[9] Margalef-Roig J., Outerelo-Domínguez E., Padrón-Fernández E. On submersions preserving the boundary and quotient manifolds, Differential Geometry and Its Aplications, World Scientific, 1990, pp. 119–128.

[10] Michor P.W., “Manifolds of Differential Mappings”, Shiva Math. Series 3, 1980.

[11] Thom R., Un lemma sur les applications differéntiables, Bol. Soc. Mat. Mex. 2 (1956), s. 1, 59–71.

S. ARMAS-GOMEZ and

E. PADRON-FERNANDEZ

Dpto. MAT. FUNDAMENTAL UNIV. de LA LAGUNA TENERIFE, CANARIAS, SPAIN E-MAIL: JSC@IAC.ES J. MARGALEF-ROIG U. MATEMATICAS C.S.I.C. SERRANO 123 28006 MADRID, SPAIN E-MAIL:VCTMO03@CC.CSIC.ES E. OUTERELO-DOMINGUEZ Dpto. GEOM.-TOPOLOGIA UNIV. COMPLUTENSE MADRID, SPAIN

Received 26 August 1992