PIECEWISE DEFINABLE C^{r}G TRIVIALITY AND DEFINABLE C^{r}G COMPACTIFICATION

TOMOHIRO KAWAKAMI

Abstract. Let G be a definably compact definable C^{r} group and 1 ≤ r < ∞. Let X be a definable C^{r}G submanifold of a representation of G and Y a definable C^{r} submanifold of R^{n}. We prove that every G invariant surjective submersive definable C^{r} map f : X → Y is piecewise definably C^{r}G trivial.

1. Introduction

Let N = (R, +, ·, <, …) be an o-minimal expansion of a real closed field R. Everything is considered in N, the term “definable” is used throughout in the sense of “definable with parameters in N ”, each definable map is assumed to be continuous and 1 ≤ r < ∞ unless otherwise stated.

General references on o-minimal structures are [2], [3], also see [12].

Definable C^{r} manifolds are studied in [11], [1], and definable C^{r}G manifolds are studied in [5], [10]. If R is the field ℝ of real numbers, then definable C^{r}G manifolds are considered in [9], [8], [7] [6].

Let f be a G invariant surjective submersive definable C^{r} map from a definable C^{r}G manifold X to a definable C^{r} manifold Y. We say that f is definably C^{r}G trivial if there exist a definable C^{r}G diffeomorphism k : X → Y × f^{−1}(a) with f = p ∘ k, where a ∈ X and p denotes the projection Y ×f^{−1}(a) → Y. We call f piecewise definably C^{r}G trivial if there exist a finite partition {C_{i}}_{i} of Y into definable C^{r} submanifolds such that each f|f^{−1}(C_{i}) is definably C^{r}G trivial.

A definable C^{r} manifold X possibly with boundary is definably compact if for every a, b ∈ R ∪ {∞} ∪ {−∞} with a < b and for every definable map f : (a, b) → X, lim_{x→a+0} f(x) and lim_{x→b−0} f(x) exist in X.

If R = ℝ, then for any definable C^{r} submanifold X of ℝ^{n}, X is compact if and only if it is definably compact. In general a definably compact definable C^{r} manifold is not necessarily compact. For example, if R = ℝ_{alg}, then [0, 1]_{ℝ_{alg}} = {x ∈ ℝ_{alg}|0 ≤ x ≤ 1} is definably compact but not compact.

Theorem 1.1. Let G be a definably compact definable C^{r} group and 1 ≤ r < ∞. Let X be a definable C^{r}G submanifold of a representation of G and Y a definable C^{r} submanifold of R^{n}. Then every G invariant surjective submersive definable C^{r} map f : X → Y is piecewise definably C^{r}G trivial.

2010 Mathematics Subject Classification. 14P10, 14P20, 57R35, 58A05, 03C64.

Keywords and P hrases. O-minimal, real closed fields, piecewise definable C^{r}G triviality, definable C^{r}G compactification.

Partially supported by Kakenhi (23540101).

If R is the field ℝ of real numbers, Theorem 1.1 is proved in [9].

A non-definably compact definable C^{r}G manifold is definably compactifiable as a definable C^{r}G manifold if it is definably C^{r}G diffeomorphic (definably G homeomorphic if r = 0) to the interior of some definably compact definable C^{r}G manifold with boundary.

Theorem 1.2. Let G be a definably compact definable C^{r} group and 2 ≤ r < ∞. Then every definable C^{r}G submanifold X of a representation Ω of G such that Ω − {0} has one orbit type and 0 ∉ X is either definably compact or definably compactifiable as a definable C^{r−1}G manifold.

If R = ℝ, then a stronger result of Theorem 1.2 is proved in [9].

In the rest of Introduction, we assume R = ℝ.

Let L > 0 and X ⊂ ℝ^{n}, Y ⊂ ℝ^{m} definable sets. A definable map f : X → Y is a definable L-Lipschitz map if for any x, y ∈ X, f satisfies the inequality ||f(x) − f(y)|| ≤ L||x − y||.

Theorem 1.3. Let G be a compact definable C^{2} group, X a definable C^{2}G submanifold of a representation of G such that X has one orbit type. Let Y a definable C^{1} submanifold of ℝ^{m}, f : X → Y a G invariant definable L-Lipschitz map, e : X → (0, ∞) a G invariant definable function and ϵ > 0. Then there exists a G invariant definable C^{1} (L+ϵ)-Lipschitz map h : X → Y such that ||h − f|| < e on X.

2. Proof of results.

Let U ⊂ R^{n}, V ⊂ R^{m} be definable open sets and f : U → V a definable map. We say that f is a definable C^{r} map if f is of class C^{r}. A definable C^{r} map is a definable C^{r} diffeomorphism if f is a C^{r} diffeomorphism.

Definition 2.1. A Hausdorff space X is an n-dimensional definable C^{r} manifold if there exist a finite open cover {U_{i}}_{i=1}^{k} of X, finite open sets {V_{i}}_{i=1}^{k} of R^{n}, and a finite collection of homeomorphisms {ϕ_{i} : U_{i} → V_{i}}_{i=1}^{k} such that for any i, j with U_{i} ∩ U_{j} ≠ ∅, ϕ_{i}(U_{i} ∩ U_{j}) is definable and ϕ_{j} ∘ ϕ_{i}^{−1} : ϕ_{i}(U_{i} ∩ U_{j}) → ϕ_{j}(U_{i} ∩ U_{j}) is a definable C^{r} diffeomorphism. This pair ({U_{i}}_{i=1}^{k}, {ϕ_{i} : U_{i} → V_{i}}_{i=1}^{k}) of sets and homeomorphisms is called a definable C^{r} coordinate system. We can define a definable C^{r} manifold with boundary.

Let G be a definable group. Let f be a G invariant surjective definable map from a definable G set X to a definable set Y. We say that f is definably G trivial if there exist a definable G homeomprphism k : X → Y × f^{−1}(a) with f = p ∘ k, where a ∈ X and p denotes the projection Y × f^{−1}(a) → Y.

By a way similar to the proof of 2.5 [9], we have the following theorem.

Theorem 2.2. Let G be a definably compact group, X a definable G set, Y a definable set and f : X → Y a G invariant definable map. Then there exists a finite partition {C_{i}}_{i} of Y into definable sets such that each f|f^{−1}(C_{i}) : f^{−1}(C_{i}) → C_{i} is definably G trivial.

By the C^{r} cell decomposition theorem (e.g. 7.3.3.2 [2]), we have the following lemma.

Lemma 2.3. Let X, Y be definable C^{r} submanifolds of R^{n}, R^{m}, respectively, and 1 ≤ r < ∞. For every definable map f : X → Y, there exists a definable open subset Z such that f|Z : Z → Y is a definable C^{r} map and dim(X − Z) < dim X.

P roof of T heorem 1.1. We proceed by induction on dim X. If dim X = 0, X is a finite set. Thus the result is clear. Assume dim X = l > 0. By Theorem 2.2, there exists a finite partition {D_{j}} of Y into definable sets such that each f|f^{−1}(D_{j}) : f^{−1}(D_{j}) → D_{j} is definably G trivial. Applying a C^{r} cell decomposition of Y compatible with {D_{j}} and replacing them, we may assume that each D_{j} is a definable C^{r} submanifold of Y.

Let X_{j} = f^{−1}(D_{j}). Then since f is G invariant and submersive, X_{j} is a definable C^{r}G submanifold of X. If dim X_{j} < l, then f|X_{j} : X_{j} → D_{j} is piecewise definably C^{r}G trivial by the inductive hypothesis. We now consider the case where dim X_{j} = l. Note that f|X_{j} : X_{j} → D_{j} is a submersion.

Since f|X_{j} : X_{j} → D_{j} is definably G trivial, there exists a definable G map h_{j} : X_{j} → F_{j} such that (f|X_{j}, h_{j}) : X_{j} → D_{j} × F_{j} is a definable G homeomorphism, where F_{j} = f^{−1}(a_{j}), a_{j} ∈ D_{j}. Note that F_{j} is a definable C^{r}G submanifold of X since f is submersive. Applying Lemma 2.3 to h_{j}, we have a G invariant definable closed subset X′_{j} of X_{j} such that dim X′_{j} < l and h_{j}|X_{j} − X′_{j} : X_{j} − X′_{j} → h_{j}(X_{j} − X′_{j}) ⊂ F_{j} is a definable C^{r}G map. Since X_{j} − X′_{j} is open and G invariant in X_{j}, f(X_{j} − X′_{j}) is a G invariant definable open subset of f(X_{j}). Hence (f, h_{j})|X_{j} −X′_{j} : X_{j} −X′_{j} → f(X_{j} −X′_{j})×h_{j}(X_{j} −X′_{j}) is a definable C^{r}G map. Applying the same argument to the inverse of (f, h_{j})|X_{j} − X′_{j}, we obtain a G invariant definable closed subset W_{j} of X_{j} − X′_{j} and a G invariant definable closed subset W′_{j} of f(X_{j} −X′_{j})×h_{j}(X_{j} −X′_{j}) such that dim W_{j}, dim W′_{j} < l and (f, h_{j})|(X_{j} −X′_{j} −W_{j}) : X_{j} − X′_{j} − W_{j} → ((f(X_{j} − X′_{j}) × h_{j}(X_{j} − X′_{j})) − W′_{j}) is a definable C^{r}G diffeomorphism. Let {U_{j}^{t}} be a C^{r} cell decomposition of X_{j} − X′_{j} − W_{j}. Since (f, h_{j})(W_{j}) = W′_{j}, each (f, h_{j})|U_{l}^{t} : U_{l}^{t} → f(U_{j}^{t}) × h_{j}(U_{l}^{t}) is a definable C^{r}G diffeomorphism.

Take a C^{r} cell decoposition {E_{k}} of f(X′_{j} ∪ W_{j}). Then each f^{−1}(E_{k}) is a definable C^{r}G submanifold of X and f|f^{−1}(E_{k}) : f^{−1}(E_{k}) → E_{k} satisfies the inductive hypothesis. Hence it is piecewise definably C^{r}G trivial. □

Theorem 2.4 ([1]). Let A be a definable closed subset of R^{n} and 0 ≤ r < ∞. Then there exists a definable C^{r} function f : R^{n} → R such that A = f^{−1}(0).

Theorem 2.5 ([10]). Let G be a definably compact definable C^{r} group, H a definable C^{r} subgroup of G, X an affine definable C^{r}G manifold and 1 ≤ r < ∞. Suppose that every orbit in X has type G/H. Then the orbit space X/G admits a unique structure of affine definable C^{r−1} manifold such that:

(1) The orbit map π : X → X/G is a definable C^{r−1} map.

(2) For any definable C^{r−1} manifold Y and a map h : X/G → Y, h is a definable C^{r−1} map if and only if so is h ∘ π.

Proposition 2.6. Let G be a definably compact definable C^{r} group, X a definable C^{r}G submanifold of a representation Ω of G such that Ω − {0} has one orbit type and 0 ∉ X and 2 ≤ r < ∞. Then X is definably C^{r−1}G imbeddable into Ω × R^{2} such that X is bounded and \overline{X} − X consists of at most one point, \overline{X} denotes the closure of X.

P roof. We may assume that X is non-definably compact. Then \overline{X} −X is a G invariant definable closed subset of Ω. Let π : Ω − {0} → (Ω − {0})/G(⊂ R^{s}) be the orbit map. Then π is definably proper. Thus π(\overline{X} − X) is a definable closed subset of R^{s}. By Theorem 2.4, there exists a definable C^{r} function f : R^{s} → R with π((\overline{X} − X) = f^{−1}(0). By Theorem 2.5, π is a definable C^{r−1} map. Thus replacing X by the graph of 1/(f ∘ π), we may assume that X is a definable C^{r−1}G submanifold of Ω × R which is closed in Ω × R. Using the stereographic projection s : Ω × R → S(Ω × R), s(X) satisfies our conditions, where S(Ω × R) denote the unit sphere of Ω × R. □

Proposition 2.7. Let X be a definable C^{r} submanifold of R^{n} and {U_{i}}_{i=1}^{l} a finite definable open cover of X and 1 ≤ r < ∞. Then there exist definable C^{r} functions λ_{1}, …, λ_{l} : X → R such that 0 ≤ λ_{i} ≤ 1, supp λ_{i} ⊂ U_{i} and ∑_{i=1}^{l} λ_{i}(x) = 1 for any x ∈ X.

We call {λ_{i}} in Proposition 2.7 a definable C^{r} partition of unity subordinate to {U_{i}}.

P roof of P roposition 2.7. As in the proof of Proposition 2.6, we may assume that X is closed in R^{n}. Hence every R^{n} − U_{i} is a definable closed subset of R^{n}. By Theorem 2.4, we have a definable C^{r} function h_{i} : R^{n} → R with h_{i}^{−1}(0) = R^{n} − U_{i}. For every i, define V_{i} = \{x ∈ X|h_{i}(x) > \frac{1}{2} max_{1≤j≤l} h_{j}(x)\}. Then \{V_{i}\}_{i=1}^{l} is a definable open cover of X and the closure \overline{V_{i}} of V_{i} in X lies in U_{i}. By Theorem 2.4, there exists a definable C^{r} function h′_{i} : R^{n} → R with h_{i}^{′−1}(0) = R^{n} − V_{i}. Hence λ_{i} := h′_{i}/ ∑_{i=1}^{l} h′_{i}, 1 ≤ i ≤ l, are the required definable C^{r} functions. □

Proposition 2.8. Let X be a definable C^{r}G submanifold closed in a representation Ω of G such that Ω − {0} has one orbit type and 0 ∉ X and {U_{i}}_{i=1}^{l} a finite G invariant definable open cover of X and 2 ≤ r < ∞. Then there exist G invariant definable C^{r−1} functions λ_{1}, …, λ_{l} : X → R such that 0 ≤ λ_{i} ≤ 1, supp λ_{i} ⊂ U_{i} and ∑_{i=1}^{l} λ_{i}(x) = 1 for any x ∈ X.

We say that {λ_{i}} in Proposition 2.8 an equivariant definable C^{r−1} partition of unity subordinate to {U_{i}}

P roof of P roposition 2.8. By Theorem 2.5, the orbit map π : Ω−{0} → (Ω−{0})/G ⊂ R^{s} is a definable C^{r−1} map. Since π|X : X → X/G is open, {π(U_{i})}_{i=1}^{l} is a finite definable open covering of a definable C^{r−1} manifold X/G. Note that π(X) is closed in R^{s} because X is closed in Ω. By Proposition 2.7, we can find a definable partition of unity \{\overline{λ_{i}}\}_{i=1}^{l} subordinate to \{π(U_{i})\}_{i=1}^{l}. Thus λ_{1} := \overline{λ_{1}} ∘ π, …, λ_{l} := \overline{λ_{l}} ∘ π are the required G invariant definable C^{r−1} functions. □

P roof of T heorem 1.2. Assume that X is non-definably compact. By Proposition 2.6, we can find a representation Ω of G and a definable C^{r−1}G imbedding i : X → Ω such that i(X) is bounded and \overline{i(X)} − i(X) = \{0\}, where \overline{i(X)} denotes the closure of X in Ω.

Let f : i(X) → R, f(x) = \frac{1}{||x||}, where ||x|| denotes the standard norm of x in Ω. By Theorem 1.1, there exist a positive element k ∈ R and a definable C^{r−1}G diffeomorphism h := (f, h_{1}) : f^{−1}((k, ∞)) → (k, ∞) × f^{−1}(k). If k is sufficiently large, then f^{−1}([0, k]) is a definably compact C^{r−1}G manifold with boundary. Hence using h and by construction of i(X) and Proposition 2.8, i(X) is definably C^{r−1}G diffeomorphic to f^{−1}([0, k)) which is the interior of f^{−1}([0, k]). □

P roof of T heorem 1.3. Since X is a definable C^{2} manifold with one orbit type and by Theorem 2.5, X/G is a definable C^{1} submanifold in some ℝ^{n} and the orbit map π : X → X/G is a definable C^{1} map. Since f, e are G invariant, they induce a definable C^{1} map \overline{f} : X/G → Y and a definable function \overline{e} : X/G → ℝ such that f = π∘\overline{f}, e = π∘\overline{e}. Since f is L-Lipschitz, \overline{f} is L′-Lipschitz for some L′ > 0. By [4], there exists a definable C^{1} (L′ + ϵ′)-Lipschitz map \overline{h} : X/G → Y such that ||\overline{h} − \overline{f}|| < \overline{e}. Therefore h = π ∘ \overline{h} is the required definable C^{1} (L + ϵ)-Lipschitz map X → Y. □

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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