DEFINABLE C^{∞}G MANIFOLD STRUCTURES OF DEFINABLE C^{r}G MANIFOLDS

TOMOHIRO KAWAKAMI

Abstract. Let G be a compact definable C^{∞} group and 2 ≤ r < ∞. Let X be a noncompact affine definable C^{r}G manifold and X_{1}, …, X_{k} noncompact codimension one definable C^{r}G submanifolds of X such that X_{1}, …, X_{k} are in general position in X and (X; X_{1}, …, X_{k}) satisfies the frontier condition. We prove that (X; X_{1}, …, X_{k}) admits a unique definable C^{∞}G manifold structure (Y; Y_{1}, …, Y_{k}).

1. Introduction^{1}

Let M = (ℝ, +.·, <, e^{x}, …) denote an exponential o-minimal expansion of the standard structure R = (R, +.·, <) of the field ℝ of real numbers such that M admits the C^{∞} cell decomposition and has piecewise controlled derivatives. The term “definable” means “definable with parameters in M”. General references on o-minimal structures are [4], [7], see also [23]. The Nash category is a special case of definable C^{∞} categories and it coincides with the definable C^{∞} category based on R [24]. Further properties and constructions of them are studied in [5], [6], [8], [21] and there are uncountably many o-minimal expansions of R [22]. Equivariant definable C^{r} cases are studied in [12], [13], [14], [15] when 0 ≤ r < ∞. Everything is considered in M and each manifold does not have boundary unless otherwise stated.

In this paper we consider simultaneous definable C^{∞}G manifold structures of definable C^{r}G manifolds and their definable C^{r}G submanifolds when 2 ≤ r < ∞.

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

Let G be a compact definable C^{r} group, X a noncompact affine definable C^{r} manifold, X_{1}, …, X_{k} noncompact definable C^{r} submanifolds of X and 1 ≤ r ≤ ∞. By 2.10 [14] and 1.3 [17], we may assume that X is a bounded definable C^{r}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.

Theorem 1.1. Let G be a compact definable C^{∞} group and 2 ≤ r < ∞. Let X be an affine definable C^{r}G manifold and X_{1}, …, X_{k} codimension one definable C^{r}G submanifolds of X such that X_{1}, …, X_{k} are in general position in X. If either X, X_{1}, …, X_{k} are compact, or X, X_{1}, …, X_{k} are noncompact and (X; X_{1}, …, X_{k}) satisfies the frontier condition, then there exist an affine definable C^{∞}G manifold Y, definable C^{∞}G submanifolds Y_{1}, …, Y_{k} of Y and a definable C^{r}G diffeomorphism f : X → Y such that for each i f(X_{i}) = Y_{i}. Moreover if Z is an affine definable C^{∞}G manifold, Z_{1}, …, Z_{k} are definable C^{∞}G submanifolds of Z and (X; X_{1}, …, X_{k}) is definably C^{r}G diffeomorphic to (Z; Z_{1}, …, Z_{k}), then (Y; Y_{1}, …, Y_{k}) is definably C^{∞}G diffeomorphic to (Z; Z_{1}, …, Z_{k}).

12000 Mathematics Subject Classification 14P10, 14P20, 58A05, 58A07, 03C64.

Keywords and Phrases. O-minimal, definable C^{∞}G manifolds, definable C^{r}G manifolds, definable C^{∞}G compactifications, raising differentiability.

A non-equivariant definable non-relative version of Theorem 1.1 is proved in 1.5 [9] and a locally definable non-relative version of it is proved in [16] when G is a finite abelian group.

Corollary 1.2. Let G be a compact definable C^{∞} group and 2 ≤ r < ∞. Let X be an affine definable C^{r}G manifold. Then X admits a unique definable C^{∞}G manifold structure up to definable C^{∞}G diffeomorphism.

If G is a finite abelian group, then every definable C^{∞}G manifold is affine [19]. By a way similar to the proof of it proves every definable C^{r}G manifold is affine when r is a non-negative integer. Thus we have the following theorem as a corollary of Theorem 1.1.

Theorem 1.3. Let G be a finite abelian group and 2 ≤ r < ∞. Let X be a definable C^{r}G manifold and codimension one X_{1}, …, X_{k} definable C^{r}G submanifolds of X such that X_{1}, …, X_{k} are in general position in X. If either X, X_{1}, …, X_{k} are compact, or X, X_{1}, …, X_{k} are noncompact and (X; X_{1}, …, X_{k}) satisfies the frontier condition, then there exist a definable C^{∞}G manifold Y, definable C^{∞}G submanifolds Y_{1}, …, Y_{k} of Y and a definable C^{r}G diffeomorphism f : X → Y such that for each i f(X_{i}) = Y_{i}. Moreover if Z is a definable C^{∞}G manifold, Z_{1}, …, Z_{k} are definable C^{∞}G submanifolds of Z and (X; X_{1}, …, X_{k}) is definably C^{r}G diffeomorphic to (Z; Z_{1}, …, Z_{k}), then (Y; Y_{1}, …, Y_{k}) is definably C^{∞}G diffeomorphic to (Z; Z_{1}, …, Z_{k}).

As a corollary of Theorem 1.3, we have the following corollary.

Corollary 1.4. Let G be a finite abelian group and 2 ≤ r < ∞. Let X be a definable C^{r}G manifold. Then X admits a unique definable C^{∞}G manifold structure up to definable C^{∞}G diffeomorphism.

2. Proof of Theorem 1.1

Suppose that r is a positive 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 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 [14].

By a way similar to the proof of 1.3 [18], we have the following theorem.

Theorem 2.1. Let G be a compact definable C^{r} group and 1 ≤ r < ∞. Let X, Y be compact affine definable C^{r}G manifolds, X_{1}, …, X_{n} (reps. Y_{1}, …, Y_{n}) compact definable C^{r}G submanifolds of X (resp Y) such that X_{1}, …, X_{n} (resp. Y_{1}, …, Y_{n}) are in general position. Suppose that f : (X; X_{1}, …, X_{n}) → (Y; Y_{1}, …, Y_{n}) is a C^{r}G map. Then f is approximated by a definable C^{r}G map h : (X; X_{1}, …, X_{n}) → (Y; Y_{1}, …, Y_{n}) in the C^{r} Whitney topology. Moreover if for 1 ≤ i_{1} < ⋯ < i_{k} ≤ n, f|X_{i_{1}}, …, f|X_{i_{k}} are definable C^{r}G maps, then we can take h such that h| ∪_{j=1}^{k} X_{i_{j}} = f| ∪_{j=1}^{k} X_{i_{j}}.

Theorem 2.2 ([10]). Let X, Y be compact C^{r} manifolds and 1 ≤ r ≤ ∞ The subset of diffeomorphisms from X to Y is open in the set of C^{r} maps from X to Y with respect to the C^{r} Whitney topology.

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. G.O. Jones [20] proved the following theorem.

Theorem 2.3 (1.2 [20]). Every definable closed subset of ℝ^{n} is the zero set of a definable C^{∞} function on ℝ^{n}.

Using Theorem 2.3, we have the following two results [17].

Theorem 2.4 ([17]). 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 2.5 ([17]). 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.

Using Theorem 2.4 and Theorem 2.5, By a way similar to the proof of 1.3 [18], we have the following theorem.

Theorem 2.6. Let G be a compact definable C^{∞} group. Let X, Y be affine definable C^{∞}G manifolds and X_{1}, …, X_{n} (reps. Y_{1}, …, Y_{n}) definable C^{∞}G submanifolds of X (resp Y) such that X_{1}, …, X_{n} (resp. Y_{1}, …, Y_{n}) are in general position. Suppose that f : (X; X_{1}, …, X_{n}) → (Y; Y_{1}, …, Y_{n}) is a definable C^{r}G map and 1 ≤ r < ∞. Then f is approximated by a definable C^{∞}G map h : (X; X_{1}, …, X_{n}) → (Y; Y_{1}, …, Y_{n}) in the definable C^{r} topology. Moreover if for 1 ≤ i_{1} < ⋯ < i_{k} ≤ n, f|X_{i_{1}}, …, f|X_{i_{k}} are definable C^{∞}G maps, then we can take h such that h| ∪_{j=1}^{k} X_{i_{j}} = f| ∪_{j=1}^{k} X_{i_{j}}.

Theorem 2.7 ([23]). Let X, Y be definable C^{r} submanifolds of ℝ^{n} and 0 < r < ∞. Let f : X → Y be a definable C^{r} map. If f is an immersion (resp. a diffeomorphism, a diffeomorphism onto its image), then an approximation of f in the definable C^{r} 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.

Let G be a compact definable C^{r} group, X a noncompact definable C^{r}G manifold, X_{1}, …, X_{k} noncompact definable C^{r}G submanifolds of X in general position in X and 1 ≤ r ≤ ∞. We say that (X; X_{1}, …, X_{k}) is simultaneously definably C^{r}G compactifiable if there exist a compact definable C^{r}G manifold Y with boundary ∂Y, compact definable C^{r}G submanifolds Y_{1}, …, Y_{k} of Y with boundary ∂Y_{1}, …, ∂Y_{n}, respectively, and a definable C^{r}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}).

To prove Theorem 1.1, we have need the following theorem.

Theorem 2.8 ([19]). Let G be a compact definable C^{r} group, X a noncompact affine definable C^{r}G manifold, X_{1}, …, X_{k} noncompact definable C^{r}G submanifolds of X in general position in X such that (X; X_{1}, …, X_{k}) satisfies the frontier condition and 1 ≤ r < ∞. Then (X; X_{1}, …, X_{k}) is simultaneously definably C^{r}G compactifiable.

In Theorem 2.8, we can take r = ∞.

Theorem 2.9 ([17]). 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.

A subset V of ℝ^{n} is an algebraic subset of ℝ^{n} if it is the zeros of some polynomial function on ℝ^{n}. An algebraic set means an algebraic subset of some ℝ^{n}. A point x in an algebraic set V ⊂ ℝ^{n} is nonsingular of dimension d in V if there exist polynomial functions p_{i} : ℝ^{n} → ℝ, (1 ≤ i ≤ n − d), and an open neighborhood U of x in ℝ^{n} such that:

(1) p_{i}(V) = 0, (1 ≤ i ≤ n − d).

(2) V ∩ U = U ∩ (∩_{i=1}^{n−d}p_{i}(0)).

(3) The gradients (∇p_{i})_{x} (1 ≤ i ≤ n − d) are linearly independent on U. The dimension dim V of V is max{d|there exists an x ∈ V which is nonsingular of dimension d}. Nonsing V = {x ∈ V |x is nonsingular of dimension dim V } and Sing V = V − Nonsing V. An algebraic set is nonsingular if Sing V = ∅. Remark that Sing V is an algebraic subset of V with dim Sing V < dim V. An algebraic subset W of a nonsingular algebraic set V is a nonsingular algebraic subset of V if W is nonsingular.

Theorem 2.10 ([1]). Let X be a compact C^{∞} manifold and X_{1}, …, X_{n} compact C^{∞} submanifolds of X in general position. Then there exist a nonsingular algebraic set Y and a C^{∞} diffeomorphism ϕ : X → Y such that each ϕ(X_{i}) is a nonsingular algebraic subset Y_{i} of Y. In particular, (X; X_{1}, …, X_{n}) admits a simultaneous Nash manifold structure (Y; Y_{1}, …, Y_{n}).

The equivariant version of Theorem 2.10 is called the relative equivariant algebraic realization problem and it is not known its complete answer. The following is its partial answer.

Theorem 2.11 ([3]). Let G be a compact definable C^{∞} group and X a compact C^{∞}G manifold. Then the disjoint union X ⨿ X of two copies of X is C^{∞}G diffeomorphic to a nonsingular algebraic G set.

Using their argument and Exercise in P58 [2], we have the following theorem.

Theorem 2.12. Let G be a compact definable C^{∞} group, X a compact C^{∞}G manifold and X_{1}, …, X_{k} codimension one compact C^{∞}G submanifolds of X in general position. Then there exist a nonsingular algebraic set Y and nonsingular algebraic G subsets Y_{1}, …, Y_{k} such that (X ⨿ X; X_{1} ⨿ X_{1}, … X_{k} ⨿ X_{k}) is C^{∞}G diffeomorphic to (Y; Y_{1}, … Y_{k}).

Corollary 2.13. Let G be a compact definable C^{∞} group, X a compact C^{∞}G manifold and X_{1}, …, X_{k} codimension one compact C^{∞}G submanifolds of X in general position. Then there exist an affine definable C^{∞}G manifold Y and definable C^{∞}G submanifolds Y_{1}, …, Y_{k} such that (X; X_{1}, … X_{k}) is C^{∞}G diffeomorphic to (Y; Y_{1}, … Y_{k}).

Using some refinement of the proof of 2.2.9 [10] and [11], we have the following theorem.

Theorem 2.14. Let G be a compact Lie group, X a compact C^{s}G manifold, X_{1}, …, X_{k} compact C^{s}G submanifolds of X in general position and 1 ≤ s < ∞. Then there exist a compact C^{∞}G manifold Y and its compact C^{∞}G submanifolds Y_{1}, …, Y_{k} of Y such that (X; X_{1}, …, X_{k}) is C^{s}G diffeomorphic to (Y; Y_{1}, …, Y_{k}).

Theorem 2.15 (1.2 [19]). Let G be a compact definable C^{r} group and 2 ≤ r < ∞. Let X be a compact affine definable C^{r}G manifold with boundary ∂X, and X_{1}, …, X_{k} compact definable C^{r}G submanifolds of X with boundary ∂X_{1}, …, ∂X_{k}, respectively, such that X_{1}, …, X_{k}, ∂X are in general position, every ∂X_{i} is contained in ∂X. Then there exists a relative definable C^{r}G collar ϕ : (∂X × [0, 1]; ∂X_{1} × [0, 1], …, ∂X_{k} × [0, 1]) → (X; X_{1}, …, X_{n}) of (∂X; ∂X_{1}, …, ∂X_{k}).

P roof of T heorem 1.1. Assume that X, X_{1}, …, X_{k} are compact. By Theorem 2.14, there exist compact C^{∞}G manifold X′ and compact C^{∞}G submanifolds X′_{1}, …, X′_{k} of X′ such that (X; X_{1}, …, X_{k}) is C^{r}G diffeomorphic to (X′; X′_{1}, …, X′_{k}). Thus by Corollary 2.13, we can find an affine definable C^{∞}G manifold Y and definable C^{∞}G submanifolds Y_{1}, …, Y_{k} of Y such that (X′; X′_{1}, …, X′_{k}) is C^{∞}G diffeomorphic to (Y; Y_{1}, …, Y_{k}). Hence (X; X_{1}, …, X_{k}) is C^{r}G diffeomorphic to (Y; Y_{1}, …, Y_{k}). By Theorem 2.1 and Theorem 2.2 and since X, X_{1}, …, X_{k} are compact, (X; X_{1}, …, X_{k}) is definably C^{r}G diffeomorphic to (Y; Y_{1}, …, Y_{k}).

Assume that X, X_{1}, …, X_{k} are noncompact and (X; X_{1}, …, X_{k}) satisfies the frontier condition. By Theorem 2.8, there exist a compact definable C^{r}G manifold X̃ with boundary ∂X̃, compact definable C^{r}G submanifolds X̃_{1}, …, X̃_{k} of X̃ with boundary ∂X̃_{1}, …, ∂X̃_{k}, respectively, and a definable C^{r}G diffeomorphism ϕ : X → Int X̃ such that ϕ(X_{i}) = Int X̃_{i}, each ∂X̃_{i} is contained in ∂X̃, and X̃_{1}, …, X̃_{k} and ∂X̃ are in general position in X̃. Thus by Theorem 2.15, (X̃; X̃_{1}, …, X̃_{k}) admits a relative definable C^{r}G collar. Hence we have the relative definable C^{r}G double (D; D_{1}, …, D_{k}) of (X̃; X̃_{1}, …, X̃_{k}). Note that D, D_{1}, …, D_{k} and ∂X̃ are compact and D_{1}, …, D_{k}, ∂X̃ are in general position.

By the argument in the first case, there exist an affine definable C^{∞}G manifold W and definable C^{∞}G submanifolds W_{1}, …, W_{k}, U of W such that (D; D_{1}, …, D_{k}, ∂X̃) is definably C^{r}G diffeomorphic to (W; W_{1}, …, W_{k}, U). Therefore we can find some unions Y, Y_{1}, …, Y_{k} of connected components of W − U, W_{1} − U, …, W_{k} − U, respectively, such that Y is an affine definable C^{∞}G manifold, each Y_{i} is a definable C^{∞}G submanifold of Y and (X; X_{1}, …, X_{k}) is definably C^{r}G diffeomorphic to (Y, Y_{1}, …, Y_{k}).

Let Z be an affine definable C^{∞}G manifold and Z_{1}, …, Z_{k} definable C^{∞}G submanifolds of Z such that (Y; Y_{1}, …, Y_{k}) is definably C^{r}G diffeomorphic to (Z; Z_{1}, …, Z_{k}) Then there exists a definable C^{r}G diffeomorphism F : (Y; Y_{1}, …, Y_{k}) → (Z; Z_{1}, …, Z_{k}). Applying Theorem 2.6 and Theorem 2.7, we have a definable C^{∞}G diffeomorphism H : (Y; Y_{1}, …, Y_{k}) → (Z; Z_{1}, …, Z_{k}). □

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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