RELATIVE DEFINABLE C^{r}G TRIVIALITY OF G INVARIANT PROPER DEFINABLE C^{r} FUNCTIONS

TOMOHIRO KAWAKAMI

Abstract. Let G be a compact definable C^{r} group and 1 ≤ r < ∞. Let X be an affine definable C^{r}G manifold and X_{1}, …, X_{k} definable C^{r}G submanifolds of X such that X_{1}, …, X_{k} are in general position in X. Suppose that f : X → ℝ is a G invariant proper surjective submersive definable C^{r} function such that for every 1 ≤ i_{1} < ⋯ < i_{s} ≤ k, f|X_{i_{1}} ∩ ⋯ ∩ X_{i_{s}} : X_{i_{1}} ∩ ⋯ ∩ X_{i_{s}} → ℝ is a proper surjective submersion. We prove that there exists a definable C^{r}G diffeomorphism h = (h′, f) : (X; X_{1}, …, X_{k}) → (X^{∗}; X_{1}^{∗}, …, X_{k}^{∗}) × ℝ, where Z^{∗}denotes Z ∩ f^{−1}(0) for a subset Z of X

Moreover we prove an equivariant definable C^{∞} version under some conditions and its application.

1. Introduction^{1}

M. Coste and M. Shiota [1] proved that a proper Nash surjective submersion f from an affine Nash manifold X to ℝ is Nash trivial, namely there exist a point a ∈ ℝ and a Nash map h : X → f^{−1}(a) such that (h, f) : X → f^{−1}(a) × ℝ is a Nash diffeomorphism.

Let M = (ℝ, +.·, <, …) denote an o-minimal expansion of the standard structure R = (R, +.·, <) of the field ℝ of real numbers. The term “definable” means “definable with parameters in M”. General references on o-minimal structures are [2], [5], see also [15]. The Nash category is a special case of the definable C^{∞} category and it coincides with the definable C^{∞} category based on R [16]. Further properties and constructions of them are studied in [3], [4], [6], [13] and there are uncountably many o-minimal expansions of R [14]. Equivariant definable C^{r} categories are studied in [7], [8], [9], [10]. Everything is considered in M and each manifold does not have boundary unless otherwise stated.

A map ψ : M → N between topological spaces is proper if for any compact set C ⊂ N, ψ^{−1}(C) is compact.

Let X be a C^{r} manifold, X_{1}, …, X_{n} C^{r} submanifolds of X and r ≥ 1. We say that {X_{i}}_{i=1}^{n} are in general position in X if for each i ∈ I and J ⊂ I − {i}, X_{i} intersects transverse to ∩_{j∈J}X_{j}.

The following is an equivariant relative definable C^{r} version of [1].

Theorem 1.1. Let G be a compact definable C^{r} group and 1 ≤ r < ∞. Let X be an affine definable C^{r}G manifold and X_{1}, …, X_{k} definable C^{r}G submanifolds of X such that X_{1}, …, X_{k} are in general position in X. Suppose that f : X → ℝ is a G invariant proper surjective submersive definable C^{r} function such that for every 1 ≤ i_{1} < ⋯ < i_{s} ≤ k, f|X_{i_{1}} ∩⋯∩X_{i_{s}} : X_{i_{1}} ∩⋯∩X_{i_{s}} → ℝ is a proper surjective submersion. Then there exists a definable C^{r}G diffeomorphism h = (h′, f) : (X; X_{1}, …, X_{k}) → (X^{∗}; X_{1}^{∗}, …, X_{k}^{∗}) × ℝ, where Z^{∗}denotes Z ∩ f^{−1}(0) for a subset Z of X.

12000 Mathematics Subject Classification 14P10, 14P20, 58A05, 03C64.

Keywords and Phrases. O-minimal, definable C^{r}G manifolds, G invariant proper definable C^{r} functions, definable C^{r}G trivial, Nash G trivial, definable C^{∞}G compactifications.

Let X = {(x, y)|y = 0} ∪ {(x, y)|xy = 1} ⊂ ℝ^{2} and f : X → ℝ, f(x, y) = x. Then f is a surjective submersive polynomial map and it is not definably trivial. Thus even in the non-equivariant category, the proper condition in Theorem 1.1 is necessary.

Let 1 ≤ r < ∞ and let F : ℝ → (−1, 1) be a definable C^{r} function such that F (x) = x in a definable open neighborhood of 0, F |(−∞, −2] = −\frac{1}{2} and F |[2, ∞) = \frac{1}{2}. Suppose that X = S^{1} × ℝ ⊂ ℝ^{3}, f : S^{1} × ℝ → ℝ, f(x, y, t) = t, X_{1} = {(0, 1)} × ℝ and X_{2} = {(x, y, t) ∈ S^{1} × ℝ|x = F (t), y = \sqrt{1 − x^{2}}\}. Then X_{1}, X_{2} are in general position in X, f, f|X_{1}, f|X_{2} are proper surjective submersions and f|X_{1} ∩ X_{2} : X_{1} ∩ X_{2} → ℝ is not surjective. Since there exists no definable C^{1} diffeomorphism h : (h′, f) : (X; X_{1}, X_{2}) → (X^{∗}; X_{1}^{∗}, X_{2}^{∗}) × ℝ, even in the non-equivariant category, the condition that every f|X_{i_{1}} ∩ ⋯ ∩ X_{i_{s}} : X_{i_{1}} ∩ ⋯ ∩ X_{i_{s}} → ℝ is a proper surjective submersion is necessary.

Let f : U → ℝ be a definable C^{∞} function on a definable open subset U ⊂ ℝ^{n}. We say that f has controlled derivatives if there exist a definable continuous function u : U → ℝ, real numbers C_{1}, C_{2}, … and natural numbers E_{1}, E_{2}, … such that |D^{α}f(x)| ≤ C_{|α|}u(x)^{E_{|α|}} for all x ∈ U and α ∈ (ℕ∪\{0\})^{n}, where D^{α} = \frac{∂^{|α|}}{∂x_{1}^{α_{1}}...∂x_{n}^{αn}} and |α| = α_{1}+⋯+α_{n}. We say that M has piecewise controlled derivatives if for every definable C^{∞} function f : U → ℝ defined in a definable open subset U of ℝ^{n}, there exist definable open sets U_{1}, …, U_{l} ⊂ U such that dim(U − ∪_{i=1}^{l}U_{i}) < n and each f|U_{i} has controlled derivatives.

The following is an equivariant definable C^{∞} version of Theorem 1.1.

Theorem 1.2. Suppose that M is exponential, admits the C^{∞} cell decomposition and has piecewise controlled derivatives. Let G be a compact definable C^{∞} group, X an affine definable C^{∞}G manifold and X_{1}, …, X_{k} definable C^{∞}G submanifolds of X such that X_{1}, …, X_{k} are in general position in X. Suppose that f : X → ℝ is a G invariant proper surjective submersive definable C^{∞} function such that for every 1 ≤ i_{1} < ⋯ < i_{s} ≤ k, f|X_{i_{1}} ∩⋯∩X_{i_{s}} : X_{i_{1}} ∩⋯∩X_{i_{s}} → ℝ is a proper surjective submersion. Then there exists a definable C^{∞}G diffeomorphism h = (h′, f) : (X; X_{1}, …, X_{k}) → (X^{∗}; X_{1}^{∗}, …, X_{k}^{∗}) × ℝ, where Z^{∗}denotes Z ∩ f^{−1}(0) for a subset Z of X.

Let G be a compact definable C^{∞} group, X a noncompact definable C^{∞}G manifold and X_{1}, …, X_{k} noncompact definable C^{∞}G submanifolds of X in general position in X. If M is exponential, admits the C^{∞} cell decomposition and has piecewise controlled derivatives and X is affine, then by Proposition 3.2, we may assume that X is a bounded definable C^{∞}G submanifold of some representation Ω of G. We say that (X; X_{1}, …, X_{k}) satisfies the frontier condition if each \overline{X_{i}} − X_{i} is contained in \overline{X} − X, where \overline{X_{i}} (resp. \overline{X}) denotes the closure of X_{i} (resp. X) in Ω. We say that (X; X_{1}, …, X_{k}) is simultaneously definably C^{∞}G compactifiable if there exist a compact definable C^{∞}G manifold Y with boundary ∂Y, compact definable C^{∞}G submanifolds Y_{1}, …, Y_{k} of Y with boundary ∂Y_{1}, …, ∂Y_{n}, respectively, and a definable C^{∞}G diffeomorphism f : X → Int Y such that for any i, f(X_{i}) = Int Y_{i}, each ∂Y_{i} is contained in ∂Y, and Y_{1}, …, Y_{k} and ∂Y are in general position in Y. Here Int Y (resp. Int Y_{i}) denotes the interior of Y (resp. Y_{i}).

As an application of Theorem 1.2, we have the following theorem.

Theorem 1.3. Suppose that M is exponential, admits the C^{∞} cell decomposition and has piecewise controlled derivatives. Let G be a compact definable C^{∞} group, X a noncompact affine definable C^{∞}G manifold and X_{1}, …, X_{k} noncompact definable C^{∞}G submanifolds of X in general position in X such that (X; X_{1}, …, X_{k}) satisfies the frontier condition. Then (X; X_{1}, …, X_{k}) is simultaneously definably C^{∞}G compactifiable.

Theorem 1.3 is an equivariant relative definable version of [1] and an equivariant definable C^{r} version is proved in [11] when r is a natural number.

2. Proof of Theorem 1.1

Let r be a non-negative integer, ∞ or ω. A definable C^{r} manifold G is a definable C^{r} group if the group operations G × G → G and G → G are definable C^{r} maps.

Let G be a definable C^{r} group. A representation map of G is a group homomorphism from G to some O_{n}(ℝ) which is a definable C^{r} map. A representation means the representation space of a representation map of G. In this paper, we assume that every representation of G is orthogonal. A definable C^{r}G submanifold of a representation Ω of G is a G invariant definable C^{r} submanifold of Ω. A definable C^{r}G manifold is a pair (X, ϕ) consisting of a definable C^{r} manifold X and a group action ϕ : G × X → X which is a definable C^{r} map. We simply write X instead of (X, ϕ). A definable C^{r}G manifold is affine if it is definably C^{r}G diffeomorphic to (definably G homeomorphic to if r = 0) a definable C^{r}G submanifold of some representation of G. Definable C^{r}G manifolds and affine definable C^{r}G manifolds are introduced in [9].

Let G be a definable C^{r} group, and X definable C^{r}G manifold and Y a definable C^{r} manifold. A G invariant definable C^{r} map f : X → Y is definably C^{r}G trivial if there exist a point y ∈ Y and a definable C^{r}G map h : X → f^{−1}(y) such that H = (h, f) : X → f^{−1}(y) × Y is a definable C^{r}G diffeomorphism.

The following is piecewise definable C^{r}G triviality of G invariant surjective submersive definable C^{r} maps [9].

Theorem 2.1 (1.1 [9]). Let r be a natural number. Let G be a compact definable C^{r} group, X an affine definable C^{r}G manifold and Y a definable C^{r} manifold. Suppose that f : X → Y is a G invariant surjective submersive definable C^{r} map. Then there exists a finite decomposition {T_{i}} of Y into definable C^{r} submanifolds of Y such that each f|f^{−1}(T_{i}) : f^{−1}(T_{i}) → T_{i} is definably C^{r}G trivial. If M admits the C^{∞} (resp. C^{ω}) cell decomposition, the we can take r = ∞ (resp. ω).

The following is existence of a definable C^{r}G tubular neighborhood of a definable C^{r}G submanifold of a representation of G.

Theorem 2.2 ([10], [8]). Let r be a non-negative integer, ∞ or ω. Then every definable C^{r}G submanifold X of a representation Ω of G has a definable C^{r}G tubular neighborhood (U, θ_{X}) of X in Ω, namely U is a G invariant definable open neighborhood of X in Ω and θ_{X} : U → X is a definable C^{r}G map with θ_{X}|X = id_{X}.

Proposition 2.3 (P4 [11]). Let r be a natural number. Let Y, Z be affine definable C^{r}G manifolds, Y_{1}, …, Y_{k} (resp. Z_{1}, …, Z_{k}) definable C^{r}G submanifolds of Y (resp. Z) in general position in Y (resp. Z). Suppose that F : (∪_{i=1}^{k}Y_{i}; Y_{1}, …, Y_{k}) → (∪_{i=1}^{k}Z_{i}; Z_{1}, …, Z_{k}) is a definable continuous G map. If each F |Y_{i} is a definable C^{r}G map (Y_{i}; Y_{i} ∩ Y_{1}, …, Y_{i} ∩ Y_{i−1}, Y_{i} ∩ Y_{i+1}, …, Y_{i} ∩ Y_{k}) → (Z_{i}; Z_{i} ∩ Z_{1}, …, Z_{i} ∩ Z_{i−1}, Z_{i} ∩ Z_{i+1}, …, Z_{i} ∩ Z_{k}), then there exist a G invariant definable open neighborhood W of ∪_{i=1}^{n}Y_{i} in Y and a definable C^{r}G map H : (W; Y_{1}, …, Y_{k}) → (Z; Z_{1}, …, Z_{k}) such that H| ∪_{i=1}^{k} Y_{i} = F.

Let 1 ≤ r < ∞ and Def^{r}(ℝ^{n}) denote the set of definable C^{r} functions on ℝ^{n}. For each f ∈ Def^{r}(ℝ^{n}) and for each positive definable continuous function ϵ : ℝ^{n} → ℝ, the ϵ-neighborhood N(f; ϵ) of f in Def^{r}(ℝ^{n}) is defined by {h ∈ Def^{r}(ℝ^{n})||D^{α}(h − f)| < ϵ, ∀α ∈ (ℕ ∪ {0})^{n}, |α| ≤ r}, where α = (α_{1}, …, α_{n}) ∈ (ℕ ∪ {0})^{n}, |α| = α_{1} + ⋯ + α_{n}. We call the topology defined by these ϵ-neighborhoods the definable C^{r} topology. By taking the relative topology of the definable C^{r} topology of ℝ^{n}, we can define the definable C^{r} topology of a definable C^{r} submanifold X of ℝ^{n}.

Let X ⊂ ℝ^{n}, Y ⊂ ℝ^{m} be definable C^{r} submanifolds. Note that if X is compact, then the definable C^{r} topology of the set of definable C^{r} maps from X to Y coincides the C^{r} Whitney topology of it [15].

Theorem 2.4 ([15]). Let X and Y be definable C^{s} submanifolds of ℝ^{n} and 0 < s < ∞. Let f : X → Y be a definable C^{s} map. If f is an immersion (resp. a diffeomorphism, a diffeomorphism onto its image), then an approximation of f in the definable C^{s} topology is an immersion (resp. a diffeomorphism, a diffeomorphism onto its image). Moreover if f is a diffeomorphism, then h^{−1} → f^{−1} as h → f.

P roof of T heorem 1.1. Since X is affine, we may assume that X is a definable C^{r}G submanifold of a representation Ω of G.

We first prove the case where k = 0. Applying Theorem 2.1, we have a partition −∞ = a_{0} < a_{1} < a_{2} < ⋯ < a_{j} < a_{j+1} = ∞ of ℝ and definable C^{r}G diffeomorphisms w_{i} : f^{−1}((a_{i}, a_{i+1})) → f^{−1}(y_{i}) × (a_{i}, a_{i+1}) with f|f^{−1}((a_{i}, a_{i+1})) = p_{i} ∘ w_{i}, 0 ≤ i ≤ j, where p_{i} denotes the projection f^{−1}(y_{i}) × (a_{i}, a_{i+1}) → (a_{i}, a_{i+1}) and y_{i} ∈ (a_{i}, a_{i+1}).

Now we prove that for each a_{i} with 1 ≤ i ≤ j, there exist an open interval I_{i} containing a_{i} and a definable C^{r}G map π_{i} : f^{−1}(I_{i}) → f^{−1}(a_{i}) such that F_{i} = (π_{i}, f) : f^{−1}(I_{i}) → f^{−1}(a_{i}) × I_{i} is a definable C^{r}G diffeomorphism. By Theorem 2.2, we have a definable C^{r}G tubular neighborhood (U_{i}, θ_{f^{−1}(a_{i})}) of f^{−1}(a_{i}) in X. Since f is proper, there exists an open interval I_{i} containing a_{i} such that f^{−1}(I_{i}) ⊂ U_{i}. Note that if f is not proper, then such an open interval does not always exist. Hence shrinking I_{i}, if necessary, F_{i} = (π_{i}, f) : f^{−1}(I_{i}) → f^{−1}(a_{i}) × I_{i} is the required definable C^{r}G diffeomorphism.

By the above argument, we have a finite family of {J_{i}}_{i=1}^{l} of open intervals and definable C^{r}G diffeomorphisms ϕ_{i} : f^{−1}(J_{i}) → f^{−1}(y_{i})×J_{i}, 1 ≤ i ≤ l, such that y_{i} ∈ J_{i}, ∪_{i=1}^{l}J_{i} = ℝ and the composition of ϕ_{i} with the projection f^{−1}(y_{i}) × J_{i} onto J_{i} is f|f^{−1}(J_{i}).

Now we glue these trivializations to get a global one. We can suppose that i ≥ 2, U_{i−1}∩J_{i} = (a, b) and ψ_{i−1} : f^{−1}(U_{i−1}) → f^{−1}(y_{1})×U_{i−1} is a definable C^{r}G diffeomorphism with f|f^{−1}(U_{i−1}) = proj_{i−1}∘ψ_{i−1}, where U_{i−1} = ∪_{s=1}^{i−1}J_{s} and proj_{i−1} denotes the projection f^{−1}(y_{1}) × U_{i−1} → U_{i−1}. Take z ∈ (a, b) = U_{i−1} ∩ J_{i}. Then since f^{−1}(y_{1}) ≅ f^{−1}(z) ≅ f^{−1}(y_{i}), f^{−1}(y_{1}) is definably C^{r}G diffeomorphic to f^{−1}(y_{i}). Hence we may assume that ϕ′_{i} is a definable C^{r}G diffeomorphism from f^{−1}(J_{i}) to f^{−1}(y_{1})×J_{i}. Then we have a definable C^{r}G diffeomorphism

ψ_{i−1} ∘ (ϕ′_{i})^{−1} : f^{−1}(y_{1}) × (a, b) → f^{−1}(y_{1}) × (a, b), (x, t) ↦ (q(x, t), t).

Take a C^{r} Nash function u : ℝ → ℝ such that u = \frac{a+b}{2} on (−∞, \frac{3}{4}a + \frac{1}{4}b] and u = id on [\frac{1}{4}a + \frac{3}{4}b, ∞). Let

τ : f^{−1}(y_{1}) × (a, b) → f^{−1}((a, b)), τ(x, t) = ψ_{i−1}^{−1}(q(x, u(t)), t).

Then τ is a definable C^{r}G diffeomorphism such that τ = (ϕ′_{i})^{−1} if \frac{1}{4}a + \frac{3}{4}b ≤ t ≤ b and τ = ψ_{i−1}^{−1} ∘ (P × id) if a ≤ t ≤ \frac{3}{4}a + \frac{1}{4}b, where P : f^{−1}(y_{1}) → f^{−1}(y_{1}), P (x) = q(x, \frac{a+b}{2}). Thus we can define

ψ̃_{i} : f^{−1}(U_{i}) → f^{−1}(y_{1}) × U_{i},

ψ̃_{i}(x) = \{ \begin{matrix}(P × id)^{−1} ∘ ψ_{i−1}(x), & f(x) ≤ \frac{3}{4}a + \frac{1}{4}b \\ τ^{−1}(x), & \frac{3}{4}a + \frac{1}{4}b ≤ f(x) ≤ b \\ ϕ_{i}(x), & f(x) > b \end{matrix} .

Then ψ̃_{i} is a definable C^{r}G diffeomorphism. Thus ψ̃_{l} : X → f^{−1}(y_{1}) × ℝ is a definable C^{r}G diffeomorphism. Therefore we have the required a definable C^{r}G diffeomorphism (H, f) : X → X^{∗} × ℝ.

We now prove the general case by induction on k.

Let k ≥ 1. By the inductive hypothesis, for any i, there exists a definable C^{r}G diffeomorphism h_{i} = (h′_{i}, f) : (X_{i}; X_{i} ∩ X_{1}, … X_{i} ∩ X_{i−1}, X_{i} ∩ X_{i+1}, …, X_{i} ∩ X_{k}) → (X_{1}^{∗}; X_{i}^{∗} ∩ X_{1}^{∗}, … X_{i}^{∗} ∩ X_{i−1}^{∗}, X_{i}^{∗} ∩ X_{i+1}^{∗}, …, X_{i}^{∗} ∩ X_{k}^{∗}) × ℝ. In particular h′_{1}|X_{2} ∩ X_{1} : (X_{2} ∩ X_{1}; X_{2} ∩ X_{1} ∩ X_{3}, …, X_{2} ∩ X_{1} ∩ X_{k}) → (X_{2}^{∗} ∩ X_{1}^{∗}; X_{2}^{∗} ∩ X_{1}^{∗} ∩ X_{3}^{∗}, …, X_{2}^{∗} ∩ X_{1}^{∗} ∩ X_{k}^{∗}) is a definable C^{r}G map. By Theorem 2.2, we have a G invariant definable open neighborhood W_{2} of X_{1} ∩ X_{2} in X_{2} and a definable C^{r}G map Φ_{2} : (W_{2}; X_{2} ∩ X_{1}; X_{2} ∩ X_{1} ∩ X_{3}, …, X_{2} ∩ X_{1} ∩ X_{k}) → (X_{2}^{∗}; X_{2}^{∗} ∩ X_{1}^{∗}; X_{2}^{∗} ∩ X_{1}^{∗} ∩ X_{3}^{∗}, …, X_{2}^{∗} ∩ X_{1}^{∗} ∩ X_{k}^{∗}) such that Φ_{2}|X_{2} ∩ X_{1} = h′_{1}|X_{2} ∩ X_{1}. Take a G invariant definable open neighborhood W′_{2} ⊂ W_{2} of X_{1} ∩X_{2} in X_{2} whose closure in X_{2} is properly contained in W_{1} and a G invariant definable C^{r} function a : X_{2} → ℝ such that its support lies in W_{2} and a|W′_{2} = 1. By Theorem 2.2, we have a G invariant definable open neighborhood O of X_{2}^{∗} in Ω and a definable C^{r}G map θ_{X_{2}^{∗}} : O → X_{2}^{∗} with θ|X_{2}^{∗} = id_{X_{2}^{∗}}.

Define

Ψ′_{2}(x) = \{ \begin{matrix}θ_{X_{2}^{∗}}((1 − a(x))h′_{2}(x) + a(x)Φ_{2}(x)), & x ∈ W_{1} \\ h′_{2}(x), & x ∈ X_{2} − W_{1} \end{matrix} .

Then Ψ′_{2} : (X_{2}; X_{2} ∩ X_{1}, … X_{2} ∩ X_{k}) → (X_{2}^{∗}; X_{2}^{∗} ∩ X_{1}^{∗}, … X_{2}^{∗} ∩ X_{k}^{∗}) is a definable C^{r}G map which is an approximation of h′_{2}. Thus h′_{1} is extensible to a definable continuous G map Ψ̃_{2} : X_{1} ∪ X_{2} → X^{∗} such that Ψ̃_{2}|X_{1} and Ψ̃_{2}|X_{2} are definable C^{r}G maps.

Repeating this process, we have a definable continuous G map Φ : (∪_{i=1}^{k}X_{i}; X_{1}, …, X_{k}) → (X^{∗}; X_{1}^{∗}, …, X_{k}^{∗}) such that each Φ|X_{i} is a definable C^{r}G map which is an approximation of h′_{i}.

By Proposition 2.3, we have a G invariant definable open neighborhood U of ∪_{i=1}^{k}X_{i} and a definable C^{r}G map L : U → X^{∗} extending Φ.

Take a G invariant definable open neighborhood U′ of ∪_{i=1}^{k}X_{i} in X whose closure in X is properly contained in U and a G invariant definable C^{r} function b : X → ℝ such that its support lies in U and b|U′ = 1. By Theorem 2.2, we have a G invariant definable open neighborhood V of X^{∗} in Ω and a definable C^{r}G map θ_{X^{∗}} : V → X^{∗} with θ_{X^{∗}}|X^{∗} = id_{X^{∗}}.

Define h′(x) =

\{ \begin{matrix}θ_{X^{∗}}((1 − b(x))H(x) + b(x)L(x)), & x ∈ U \\ H(x), & x ∈ X − U \end{matrix} .

Then h′ : (X; X_{1}, …, X_{k}) → (X^{∗}; X_{1}^{∗}, …, X_{k}^{∗}) is a definable C^{r}G map. Thus h = (h′, f) : (X; X_{1}, …, X_{k}) → (X^{∗}; X_{1}^{∗}, …, X_{k}^{∗}) × ℝ is a definable C^{r}G map which is an approximation of (H, f). Therefore by Theorem 2.4, h is the required definable C^{r}G diffeomorphism. □

3. Proof of Theorem 1.2 and 1.3

From now on we assume that M is exponential, admits the C^{∞} cell decomposition and has piecewise controlled derivatives.

Theorem 3.1 (1.2 [12]). Every definable closed subset of ℝ^{n} is the zero set of a definable C^{∞} function on ℝ^{n}.

Proposition 3.2. Let G be a compact definable C^{∞} group and X a definable C^{∞}G manifold in a representation Ω of G. Then X is definably C^{∞}G imbeddable into Ω × ℝ^{2} such that X is bounded and \overline{X} − X consists of at most one point, where \overline{X} denotes the closure of X.

P roof. We may assume that X is noncompact. Then \overline{X} − X is a G invariant closed definable subset of Ω. Let π : Ω → Ω/G ⊂ ℝ^{s} denote the orbit map. Then i∘π : Ω → ℝ^{s} is a proper polynomial map (see Section 4 [10]), where i : Ω/G → ℝ^{s} denotes the inclusion. Hence i∘π|\overline{X} −X : \overline{X} −X → ℝ^{s} is proper because \overline{X} −X is closed in Ω. Thus i∘π(\overline{X} −X) (=π(\overline{X} − X)) is a definable closed subset of ℝ^{s}. Applying Theorem 3.1, there exists a definable C^{∞} function f : ℝ^{s} → ℝ with π(\overline{X}−X) = f^{−1}(0). Hence F := f ∘π : Ω → ℝ is a G invariant definable C^{∞} function with \overline{X} − X = F^{−1}(0). Therefore replacing the graph of 1/F by X, we may assume that X is closed in Ω × ℝ. Applying the stereographic projection s : Ω × ℝ → S(Ω × ℝ^{2}), s(X) satisfies our requirements, where S(Ω × ℝ^{2}) denotes the unit sphere of Ω × ℝ^{2}. □

The proof of Proposition 3.2 proves the following two theorems and proposition.

Theorem 3.3. Let G be a compact definable C^{∞} group and Ω a representation of G. Every G invariant definable closed subset of Ω is the zero set of a G invariant definable C^{∞} function on Ω.

Theorem 3.4. Let G be a compact definable C^{∞} group and X an affine definable C^{∞}G manifold. Suppose that A, B are G invariant definable disjoint closed subsets of X. Then there exists a G invariant definable C^{∞} function f : X → ℝ such that f|A = 1 and f|B = 0.

Proposition 3.5. Let G be a compact definable C^{∞} group, X a noncompact affine definable C^{∞}G manifold and X_{1}, …, X_{n} noncompact definable C^{r}G submanifolds of X in general position in X such that (X; X_{1}, …, X_{n}) satisfies the frontier condition. Then we may assume that X is a bounded definable C^{∞}G submanifold of some representation Ω of G such that \overline{X_{1}} − X_{1} = ⋯ = \overline{X_{n}} − X_{n} = \overline{X} − X = \{0\}, where \overline{X} (resp. \overline{X_{i}}) denotes the closure of X (resp. X_{i}) in Ω.

Using Theorem 3.4, a similar proof of P4 [11] proves the following proposition.

Proposition 3.6. Let Y, Z be affine definable C^{∞}G manifolds, Y_{1}, …, Y_{k} (resp. Z_{1}, …, Z_{k}) definable C^{∞}G submanifolds of Y (resp. Z) in general position in Y (resp. Z). Suppose that F : (∪_{i=1}^{k}Y_{i}; Y_{1}, …, Y_{k}) → (∪_{i=1}^{k}Z_{i}; Z_{1}, …, Z_{k}) is a definable continuous G map. If each F |Y_{i} is a definable C^{∞}G map (Y_{i}; Y_{i}∩Y_{1}, …, Y_{i}∩Y_{i−1}, Y_{i}∩Y_{i+1}, …, Y_{i}∩Y_{k}) → (Z_{i}; Z_{i} ∩ Z_{1}, …, Z_{i} ∩ Z_{i−1}, Z_{i} ∩ Z_{i+1}, …, Z_{i} ∩ Z_{k}), then there exist a G invariant definable open neighborhood W of ∪_{i=1}^{n}Y_{i} in Y and a definable C^{∞}G map H : (W; Y_{1}, …, Y_{k}) → (Z; Z_{1}, …, Z_{k}) such that H| ∪_{i=1}^{k} Y_{i} = F.

P roof of T heorem 1.2. Using Theorem 3.4 and Proposition 3.6, a similar proof of Theorem 1.1 proves Theorem 1.2. □

P roof of T heorem 1.3. By Proposition 3.5, we may assume that X is a bounded definable C^{∞}G submanifold of a representation Ω of G such that \overline{X_{1}} − X_{1} = ⋯ = \overline{X_{n}} − X_{n} = \overline{X} − X = \{0\}.

Let f : X → ℝ, f(x) = ||x||^{−1}, where ||x|| denotes the standard norm of x in Ω. Since f is submersive and G invariant and by Theorem 2.1, there exist a sufficiently large positive number α and a definable C^{∞}G map h_{1} : f^{−1}((α, ∞)) → f^{−1}(α) such that h := (h_{1}, f) : f^{−1}((α, ∞)) → f^{−1}(α) × (α, ∞) is a definable C^{∞}G diffeomorphism.

Let f_{i} := f|X_{i}. Since (X; X_{1}, …, X_{k}) satisfies the frontier condition and X_{1}, …, X_{k} are in general position in X, each Y_{i} := f_{i}^{−1}((α, ∞)) is a definable C^{∞}G submanifold of Y := f^{−1}((α, ∞)), Y_{1}, …, Y_{k} are in general position in Y and for every 1 ≤ i_{1} < ⋯ < i_{s} ≤ k, f|Y_{i_{1}} ∩ ⋯ ∩ Y_{i_{s}} : Y_{i_{1}} ∩ ⋯ ∩ Y_{i_{s}} → (α, ∞) is a proper surjective submersion. Since (α, ∞) is definably C^{∞} diffeomorphic to ℝ, there exists a G invariant surjective submersive definable C^{∞} function F : (Y; Y_{1}, …, Y_{k}) → ℝ satisfying the conditions in Theorem 1.2.

Applying Theorem 1.2 to F, there exists a definable C^{∞}G diffeomorphism (f^{−1}((α, ∞)); f_{1}^{−1}((α, ∞)), …, f_{k}^{−1}((α, ∞))) → (f^{−1}(α); f_{1}^{−1}(α), …, f_{k}^{−1}(α))×ℝ. Thus we have a definable C^{∞}G diffeomorphism H : (f^{−1}((α, ∞)); f_{1}^{−1}((α, ∞)), …, f_{k}^{−1}((α, ∞))) → (f^{−1}(α); f_{1}^{−1}(α), …, f_{k}^{−1}(α)) × (α, ∞). Since α is sufficiently large, f^{−1}([0, α + 1]) is a compact definable C^{∞}G manifold with boundary f^{−1}(α + 1) and each f_{i}^{−1}([0, α + 1]) is a compact definable C^{∞}G submanifold of f^{−1}([0, α + 1]) with boundary f_{i}^{−1}(α + 1). Therefore using H and Theorem 3.4, (X; X_{1}, …, X_{k}) is definably C^{∞}G diffeomorphic to (f^{−1}([0, α + 1); f_{1}^{−1}([0, α + 1), …, f_{k}^{−1}([0, α + 1))). □

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Department of Mathematics, Faculty of Education, Wakayama University, Sakaedani Wakayama 640-8510, Japan

E-mail address: kawa@center.wakayama-u.ac.jp