APPROXIMATION THEOREMS IN O-MINIMAL ... - Project Euclid

Illinois Journal of Mathematics Volume 46, Number 1, Spring 2002, Pages 111–128 S 0019-2082

APPROXIMATION THEOREMS IN O-MINIMAL STRUCTURES ´ ESCRIBANO JESUS Abstract. We prove that, in any o-minimal structure, definable C r mappings can always be approximated by definable C r+1 mappings. As an application we obtain definable triviality for pairs of definable proper submersions.

Introduction In this note we study the following interesting problem in Differential Topology: the approximation of C r mappings by C s mappings, where r < s < ∞. We study this problem in an o-minimal context, that is, we work on C r mappings definable in an o-minimal structure expanding a real closed field. Classical arguments for this problem (for example, convolutions; see [9]) cannot be used here, as the integration of definable functions does not necessarily produce definable functions. Using this approximation result, we can extend to the o-minimal context the triviality results for submersions proved by Coste and Shiota [4] and by the author [7] in the semialgebraic case. Let us recall the basic notions involved. A structure expanding a real closed field R is a collection S = (Sn )n∈N , where each Sn is a Boolean subalgebra of subsets of the affine space Rn that contains all algebraic sets of Rn and such that A × B ∈ Sm+n if A ∈ Sm and B ∈ Sn , and π(A) ∈ Sn if π : Rn+1 → Rn is the projection on the first n coordinates and A ∈ Sn+1 . The elements of Sn are called the definable subsets of Rn . The structure S is said to be o-minimal if the elements of S1 are precisely the finite unions of points and intervals. Of course, the first model for an o-minimal geometry is semialgebraic geometry. Nice references on the subject are [3] and [5]. In the following, we shall always work in an o-minimal structure expanding a real closed field R. It is easy to translate to an arbitrary real closed field R the usual notions of differentiability over R. Basic results on differentiability for semialgebraic Received February 2, 2001; received in final form December 7, 2001. 2000 Mathematics Subject Classification. Primary 03C64. Secondary 57R12. Partially supported by DGICYT, PB98-0756-C02-01. c

2002 University of Illinois

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functions defined over R can be found in [1]. Van den Dries’ work [5] gives results of this kind for definable functions. We use the following notation. If a C k function f : U → R, where U ⊂ Rn is an open definable subset, is definable in an o-minimal structure S, we say that f is (of class) DSk (or simply Dk when there is no ambiguity about S). In other words, Dk means “definable of class C k ”. In [10], the notion of a Dr manifold is introduced. We can construct the usual objects attached to a Dr manifold, such as tangent and normal bundles (see [3] and [10]). We establish our results for r < ∞ because the D∞ category is not well behaved (see [13]). Our approximation theorem replaces the approximation theorem for the Nash case established by Shiota [11] and allows us to obtain the equivalence of the Dr and Dr+1 categories. This can be applied, for instance, to construct tubular neighbourhoods without loss in the order of differentiability. We remark here that [12, Sec. II.6] contains results of this type, although the proofs given there are quite different and difficult to follow, even in the case of the reals. In Section 1, we state and prove the approximation theorem for Dr mappings. We prove the results in several steps, obtaining at each step a partial approximation result. One of these steps is a result that is of great interest in itself: it asserts the existence of Dr tubular neighbourhoods for Dr manifolds. We apply these results to prove a result (Theorem 1.10) about the smoothing of definable corners. In Section 2 we explain how the triviality of pairs of proper definable submersions follows, once our approximation theorem is available. To prove the triviality results, we introduce the concepts of the definable spectrum and an elementary extension. This paper is based on our Ph.D. thesis [6]. We wish to thank our advisors Michel Coste and Jes´ us M. Ruiz for their help and support. 1. Approximation theorems in o-minimal structures In this section we prove the following approximation theorem for definable mappings in an o-minimal structure: Theorem 1.1. Given two Dk manifolds X ⊂ Rn and Y ⊂ Rm , each Dk−1 mapping f : X → Y admits a Dk approximation fe : X → Y .

(This result corresponds to [6, Th. 4.7.1].) To make the statement of the theorem precise, we must first define a topology in the spaces of Dr mappings, for r < ∞. Once this is done, we will prove several partial results, which will then lead us to the proof of the main theorem. Let S = (Sn )n∈N be an o-minimal structure expanding the real closed field R. Let X and Y be Dr submanifolds of Rn and Rm , respectively. Fix k ≤ r.

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We denote by DSk (X, Y ) the set of Dk mappings X → Y . We write Dk (X, Y ) when there is no confusion about S, and Dk (X) when Y = R. To define a topology on Dk (X, Y ), consider first the case Y = R. We use the following notation. If V is a Dk−1 vector field on X and f ∈ Dk (X), we write V f for the derivative of f along V , that is, V f (x) = Df (x)(V (x)) for each x ∈ X. We take a finite family {V1 , . . . , Vp } of Dk−1 vector fields on X that span the tangent space to X at each point of X, that is, hV1 (x), . . . , Vp (x)i = Tx X for each x ∈ X. For each definable continuous positive function ε on X let Uε = g ∈ Dk (X) : |Vi1 · · · Vij g| < ε for 1 ≤ i1 , . . . , ik ≤ p, j ≤ k .

Then the sets {h + Uε }ε form a neighbourhood basis of Dk (X) at h, which defines a topology on Dk (X). The topology on Dk (X, Rm ) = Dk (X, R) × · · · × Dk (X, R) is simply the product topology. For a manifold Y ⊂ Rm , Dk (X, Y ) is a subset of Dk (X, Rm ), and we can restrict to Dk (X, Y ) the topology of Dk (X, Rm ). We can always choose a finite set {V1 , . . . , Vp } of vector fields on X that verifies the above condition (specifically, we can take the projections on the tangent spaces Tx X of the vector fields ∂/∂xi , where x1 , . . . , xn are the coordinates in Rn ), and it is easy to check that the topology does not depend on the choice of {V1 , . . . , Vp }. We will call this topology the Dk topology. It is a definable version of the strong Whitney topology. Proposition 1.2. Given a Dr submanifold X ⊂ Rn and a closed Dr submanifold Y ⊂ X, the restriction mapping res : Dr (X) → Dr (Y ) : f 7→ f|Y is continuous for the Dk topology. Proof. (See [6, Prop. 4.1.2].) Consider a set {V1 , . . . , Vq } of Dr−1 vector fields on X such that {V1 (x), . . . , Vq (x)} generates Tx X at each x ∈ X. Reordering the Vi ’s if necessary, we can assume that there exists p ≤ q such that {V1 (x), . . . , Vp (x)} generates Tx Y for each x ∈ Y . By this construction, if we take a neighbourhood Uε of 0 as above, we can construct a definable function ε : X → R such that 0 ∈ Uε ⊂ res−1 (Uε ). To construct ε, let T be an open definable tubular neighbourhood of Y in X, and π : T → Y a Dr−1 retraction (see Remark 1.4 below). Take a Dr partition of unity {θ, 1 − θ} of X subordinated to the covering {T, X \ Y }, and set ε = θ(ε ◦ π) + (1 − θ). If Y is not closed, we obtain the continuity of res : Dr (U ) → Dr (Y )

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in an open definable neighbourhood U of Y in X (we can take U = X \Bd(Y )). A main property of this topology is that the Dk diffeomorphisms form an open subset of Dk (X, Y ). (To see this, it is enough to repeat the argument given in [11, Lemma II.1.7] for the semialgebraic case.) Moreover, the Dk embeddings form an open subset. Another easy, but useful result is the following: Proposition 1.3. Let X ⊂ Rm , Y ⊂ Rn and Z ⊂ Rp be Dk manifolds. Let h : Y → Z be a Dk mapping. Then the mapping h∗ : Dk (X, Y ) → Dk (X, Z) f 7→ h∗ (f ) = h ◦ f is continuous (for the D topology). k

Proof. See [6, Prop. 4.1.3].

For our approximation results, we will need the following construction. Remark 1.4. Let M be a Dr submanifold of Rn . The usual normal bundle is a Dr−1 manifold, and there exists a Dr−1 diffeomorphism from a definable open neighbourhood of the zero section M × {0} onto a definable open neighbourhood Ω of M in Rn . This neighbourhood Ω is called a Dr−1 tubular neighbourhood of M , and as usual we have a Dr−1 retraction π : Ω → M and a Dr−1 “square of the distance function” ρ : Ω → R. (See [3, Th. 6.11] or [12, Lemma II.5.1] for details. In the latter reference, the result is established in a somewhat different setting.) Having defined the topology and our auxiliary construction, we now turn to approximations. We begin by proving an approximation theorem under some additional restrictions. We fix the following notation. We write (x, y) = (x1 , . . . , xm , y1 , . . . , yn ) ∈ Rm+n . Given a = (a1 , . . . , am ) ∈ Nm and b = (b1 , . . . , bn ) ∈ Nn , we set |a| = a1 + · · · + am , |b| = b1 + · · · + bn and ∂ |a|+|b| f ∂ |a|+|b| f = . a b a ∂x ∂y ∂x 1 · · · ∂xam ∂y b1 · · · ∂y bn Theorem 1.5. Let f : Rm+n → R be a Dr function. Assume that f is D off {0} × Rn and that the mappings r+1

∂ |a| f (0, y) ∂xa are Dr+1 for each a ∈ Nm , 0 ≤ |a| ≤ r. Let δ : Rm+n → R be a positive continuous definable function. Then there exists a Dr+1 function fe : Rm+n → R such that |a|+|b| ∂ e) < δ (f − f ∂xa ∂y b Rn → R : y 7→


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for 0 ≤ |a| + |b| ≤ r. Proof. By Taylor’s formula, we have f (x, y) = f (0, y) +

m X X ∂f (0, y)xi + · · · + ∂xi i=1

|a|=r−1

+

1 ∂ r−1 f (0, y)xa a! ∂xa

X 1 ∂rf (ξ, y)xa a! ∂xa

|a|=r

for a suitable ξ in the segment [0, x]. We define θ(x, y) = f (0, y) +

m X X 1 ∂rf ∂f (0, y)xi + · · · + (0, y)xa a ∂x a! ∂x i i=1 |a|=r

and Q(x, z, y) =

X 1 ∂rf ∂rf (z, y) − (0, y) xa , a! ∂xa ∂xa

|a|=r

so that for each x and y there exists ξ ∈ [0, x] such that f (x, y) = θ(x, y) + Q(x, ξ, y). Note that Q is a polynomial in x whose coefficients are the continuous functions 1 ∂rf ∂rf Φa (z, y) = (z, y) − a (0, y) . a! ∂xa ∂x Consider the set A = {(x, y, ξ) ∈ Rm × Rn × Rm : f (x, y) = θ(x, y) + Q(x, ξ, y), ξ ∈ [0, x]} . This set is definable, and for every x ∈ Rm and y ∈ Rn there exists ξ such that (x, y, ξ) belongs to A. Hence, by Definable Choice [5, Prop. 6.1.2] there exists a definable function ζ : Rm × Rn → Rm such that for each x ∈ Rm and y ∈ Rn , (x, y, ζ(x, y)) ∈ A. Moreover, as ξ ∈ [0, x], we see that ζ(0, y) = 0 and ζ(x, y) → 0 as (x, y) → (0, y0 ). Hence X 1 f (x, y) = θ(x, y) + χa (x, y)xa , χa (x, y) = Φa (ζ(x, y), y), a! |a|=r

where the χa ’s are definable functions such that χa (0, y) = 0 and χa (x, y) → 0 as (x, y) → (0, y0 ). We set X 1 Q(x, y) = χa (x, y)xa . a! |a|=r

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Let 1 : R → R be a sufficiently small positive Dr+1 function and define : Rn → R by (y) = 1 (|y|2 ). We can choose 1 such that the derivatives X ∂ |a| (|a|−|q|) = Ka,q 1 (|y|2 )y a−2q , ∂y a q≤[a/2]

where the Ka,q ’s are constants, and [a/2] is the integer part of a/2, are bounded by constants. Consider the set U = (x, y) ∈ Rm+n : |x| < (y) . We define

|x|2 , 2 (y) where µ : R → R is a Dr+1 function such that µ ≡ 1 in a neighbourhood of 0 and µ ≡ 0 outside the interval [−1, 1]. Finally, we set λ(x, y) = µ

fe(x, y) = θ(x, y) + (1 − λ(x, y))Q(x, y).

The function fe is Dr+1 off {0} × Rn because all functions used in its definition are Dr+1 . Moreover, in a neighbourhood of {0} × Rn , fe is equal to θ, and hence is Dr+1 . Thus, fe is Dr+1 . We next show that fe is a good approximation. As fe = f off U , it is enough to consider U . We first observe that |f (x, y) − fe(x, y)| = |λ(x, y)||Q(x, y)| ≤ |Q(x, y)|. But since (x, y) ∈ U and Q(0, y) = 0, we can make |Q(x, y)| arbitrary small by taking small enough. We next consider the derivatives. Using the above notation, we have ∂ |p|+|q| λ ∂ |a−p|+|b−q| Q ∂ |a|+|b| (f − fe) X X = A , p,q ∂xa ∂y b ∂xp ∂y q ∂xa−p ∂y b−q p≤a q≤b

where p = (p1 , . . . , pm ), q = (q1 , . . . , qn ) and the Ap,q ’s are constants. By induction one proves easily that |a| X |a|−2|q| ∂ |a| λ 1 |x|2 1 (|a|−|q|) a−2q = C µ x , a,q ∂xa (y) 2 (y) (y) q≤[a/2]

where the Ca,q ’s are constants. |a|+|b| Hence, to study ∂∂xa ∂ybλ we must analyze the derivatives with respect to y of expressions of the form 2γ |x|2 1 (γ) µ . 2 (y) (y) Again by induction, one sees that each term ∂ |k| |x|2 (γ) µ ∂y k 2 (y)


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is a sum of terms of the form 2|s|+|α|+|β| 1 |x|2 (γ+|s|) 2|s| |x| µ Φα,β,s , 2 (y) (y) where αi + βi ≤ ki , 1 ≤ si ≤ αi , and Φα,β,s is a polynomial in and its derivatives. On the other hand, ∂ |k| 1 1 = Λk , ∂y k (y)a (y)a+k where Λk is a polynomial in and its derivatives. The derivatives of Q are X ∂ |a+b| Q = δq (x, y)xq , a b ∂x ∂y |q|=r−(|a|+|b|)

where δq is a definable function such that δ(0, y) = 0 and δ(x, y) → 0 when (x, y) → (0, y0 ). Combining these calculations, we see that each term ∂ |p+q| λ ∂ |a−p|+|b−q| Q ∂xp ∂y q ∂xa−p ∂y b−q is a sum of terms of the form 2|s|+|α|+|β| |x|2 1 µ(|p|−|ν|+|s|) 2 |x|2|s| xp−2ν · (y) (y) 2|p|−2|ν|+|q|−|k| 1 Ξk,α,β,s (y)δt (x, y)xt , · (y) where νi ≤ [pi /2], ki ≤ qi , αi + βi ≤ ki , 1 ≤ si ≤ αi , |t| = r − (|a − p| + |b − q|), and Ξk,α,β,s is a polynomial in and its derivatives. To bound the above expression, it is enough to bound 2|s|+|α|+|β| 2|p|−2|ν|+|q|−|k| 1 1 2|s| p−2ν |x| x xt . (y) (y) But since we assumed that |x| < (y), this expression is bounded by (y)2|s| (y)|p|−2|ν|

1 1 (y)|t| (y)2|s|+|α|+|β| (y)2|p|−2|ν|+|q|−|k|

(y)|p| (y)r−|a|+|p|−|b|+|q| = (y)r−(|a|+|b|) (y)|k|−(|α|+|β|) , (y)|α|+|β| (y)2|p|+|q|−|k| and it is easily seen that the last expression is bounded. =

We now prove an approximation theorem for arbitrary functions on Rn , without any restrictions.

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Theorem 1.6. Let f : Rn → R be a Dr function. Let δ : Rn → R be a positive continuous definable function. There exists a Dr+1 approximation fe : Rn → R of f such that |α| ∂ e ∂xα (f − f ) < δ for 0 ≤ |α| ≤ r.

S Proof. We first choose a finite stratification Rn = Mi such that, for each i, Mi is a Dp cell, Mi is Dp diffeomorphic to Rdi (where di = dim Mi ), and f|Mi : Mi → R is a Dp function, for an integer p that is large enough so that the conditions stated below hold (see [5, Ch. 7.3]). For each i, Mi has a Dp−1 tubular neighbourhood (see [3, Th. 6.11]), that is, a definable open neighbourhood Ti of Mi in Rn , a Dp−1 submersive retraction τi : Ti → Mi and a Dp−1 “square of distance to Mi ” function ρi : Ti → R. We can also assume that, for each i, there is a Dp−1 diffeomorphism φi : Ti → Rn such that φi (Mi ) = {0} × Rdi . (Note that the tubular neighbourhood is trivial because Mi is Dp diffeomorphic to an affine space.) We define [ M ≤k = Mi . codim(Mi )≤k

As {Mi } is a stratification, for each k = 0, . . . , n, the union of strata of dimension < n − k is a closed subset of Rn , and hence its complement M ≤k is an open subset of Rn . In this situation, we will prove by induction on k that there exists a Dr+1 function fe≤k : M ≤k → R such that |α| ∂ ≤k e ∂xα (f − f )(x) ≤ ηk (x)

for each x ∈ M ≤k , where ηk : Rn → R is a nonnegative continuous definable function such that ηk < δ and ηk ≡ 0 on Rn \M ≤k . In the case k = 0, M ≤0 is a union of open definable subsets of Rn , so we can just take fe≤0 = f and η0 = 0. Assume k > 0. By induction, there is a Dr+1 function fe