REPRESENTATIVE DEFINABLE C^{r} FUNCTIONS ON DEFINABLE C^{r} GROUPS

TOMOHIRO KAWAKAMI

Abstract. Let G be a compact affine definable C^{r} group and let r be ∞ or ω. We prove that the representative definable C^{r} functions on G is dense in the space of continuous functions on G.

1. Introduction.

Let M = (ℝ, +, ·, <, …) be an o-minimal expansion of the standard structure R = (ℝ, +, ·, <) of the field ℝ of real numbers. Everything is considered in M, every definable map is assumed to be continuous and the term “definable” is used throughout in the sense of “definable with parameters in N ”unless otherwise stated. We assume that r denotes ∞ or ω.

General references on o-minimal structures are [1], [2], also see [13].

Definable C^{r}G manifolds and definable G sets in M are studied in [8], [7], [6].

Let G be a definable C^{r} group and Def^{r}(G) denote the space of definable C^{r} functions. Left translations in G induce an action of G defined by f : G → ℝ ↦ L(g, f) = f(g^{−1}x) : G → ℝ. A function f on G is representative if the functions {L(g, f)|g ∈ G} generate a finite dimensional subspace of Def^{r}(G).

Theorem 1.1. Let G be a compact affine definable C^{r} group. Then the representative definable C^{r} functions on G is dense in the strong topology in the space of continuous functions on G.

Let X be a definable C^{r}G manifold. We say that the action of G on X is definably C^{r} linearizable (resp. C^{r} linearizable) if there exist a definable C^{r} representation of G whose representation space Ω, a definable C^{r}G submanifold Y of Ω and a definable C^{r}G diffeomorphism (resp. C^{r} diffeomorphism) from X to Y.

Theorem 1.2. Let G be a compact affine definable C^{r} group and X a compact definable C^{r}G manifold. Then the action is C^{r} linearizable.

Remark that if M = R, then for any positive dimensional compact connected C^{∞}G manifold with non-transitive action, it admits uncountably many nonaffine definable C^{∞}G manifold structures ([10]). In Theorem 1.2, we cannot replace C^{r} linearizable by definably C^{r} linearizable.

Locally definable C^{r} manifolds are defined in [9].

2010 Mathematics Subject Classification. 57S15, 03C64.

Keywords and P hrases. O-minimal, definable C^{r} groups, definable C^{r}G manifolds.

Partially supported by Kakenhi (23540101)

Theorem 1.3. Let G be a connected locally definable C^{r} group and (G̃, π) the universal cover of G. Then G̃ can be equipped uniquely with the structure of a locally definable C^{r} group such that π is a locally definable C^{r} group homomorphism.

A locally Nash case of Theorem 1.3 is proved in [5].

2. Preliminaries and proof of results

Let X ⊂ ℝ^{n} and Y ⊂ ℝ^{m} be definable sets. A continuous map f : X → Y is definable if the graph of f (⊂ X × Y ⊂ ℝ^{n} × ℝ^{m}) is a definable set.

We say that a group G is a definable group if G is a definable set and the group operations G × G → G and G → G are definable.

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. A definable C^{r} manifold X is affine if X is definably C^{r} diffeomorphic to a definable C^{r} submanifold of some ℝ^{n}.

A definable C^{r} manifold (resp. An affine definable C^{r} manifold) G is a definable C^{r} group (resp. an affine definable C^{r} group) if G is a group and the group operations G × G → G, G → G are definable C^{r} maps.

A subgroup of a definable C^{r} group is a definable subgroup of it if it is a definable C^{r} submanifold of it. Note that every definable C^{r} subgroup of a definable C^{r} group is closed ([12]) and a closed subgroup of a definable C^{r} group is not necessarily definable.

Let G be a definable C^{r} group. A group homomorphism from G to some O_{n}(ℝ) is a definable C^{r} representation if it is a definable C^{r} map. A definable C^{r} representation space of G is ℝ^{n} with the orthogonal action induced from a definable C^{r} representation of G. A definable C^{r}G submanifold means a G invariant definable C^{r} submanifold of some definable C^{r} representation space of G.

Let G be a definable C^{r} group. A defin-able C^{r}G manifold is a pair (X, ϕ) consisting of a definable C^{r} manifold X and a definable C^{r} action ϕ : G × X → X on X of G. For abbreviation, we 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 definable C^{r} representation space of G.

P roof of T heorem 1.1. Since G is compact and affine, there exists a definable C^{r}G diffeomorphism f from G to a definable C^{r}G submanifold G′ of some definable C^{r} representation space Ω of G.

Let r : G → ℝ be a continuous function. Applying Polynomial Approximation Theorem to r ∘ f^{−1} : G′ → ℝ, we have a polynomial function q : G′ → ℝ approximating r ∘ f^{−1}. Since f is equivariant and G acts orthogonally on Ω and by P107 [11], q ∘ f : G → ℝ is a representative on G which is a definable C^{r} function approximating r. □

By a way similar to the proof of results of [10], we have the following result.

Theorem 2.1. Let G be a compact affine definable C^{r} group and X a compact C^{∞}G manifold. Then X is C^{∞}G diffeomorphic to a definable C^{r}G submanifold Y of some representation space of G.

P roof of T heorem 1.2. We only have to prove the case where r = ω. By Theorem 2.1, there exist a representation space Ω of a definable C^{r} representation of G, a definable C^{r}G submanifold Y of Ω and a C^{∞}G diffeomorphism f : X → Y. By [P 233 [4]], any Whitney neighborhood of a C^{∞}G map to a representation space contains a C^{ω}G map. Thus we can approximate f by a C^{ω}G map h : X → Ω. Therefore we have a required C^{ω}G imbedding. □

A Hausdorff space X is an n-dimensional locally definable C^{r} manifold if there exist a countable open cover {U_{i}}_{i=1}^{∞} of X, countably many open sets {V_{i}}_{i=1}^{∞} of ℝ^{n}, and a countable collection of homeomorphisms {ϕ_{i} : U_{i} → V_{i}}_{i=1}^{∞} 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. We call the (U_{i}, ϕ_{i})′s the definable charts of X.

Note that locally definable (C^{0}) manifolds are considered in [3].

Let X, Y be locally definable C^{r} manifolds with definable charts (U_{i}, ϕ_{i})_{i∈I}, (W_{j}, ψ_{j})_{j∈J} respectively. A continuous map f : X → Y is a locally definable C^{r} map if for every finite subset I′ of I, there exists a finite subset J′ of J such that f(∪_{i∈I′}U_{i}) ⊂ ∪_{j∈J′}V_{j} and that f| ∪_{i∈I′} U_{i} : ∪_{i∈I′}U_{i} → ∪_{j∈J′}V_{j} is a definable C^{r} map.

A bijective locally definable C^{r} map f between locally definable C^{r} manifolds is a locally definable C^{r} diffeomorphism if f^{−1} is a locally definable C^{r} map.

A locally definable C^{r} manifold X is affine if X is locally definably C^{r} diffeomorphic to a locally definable C^{r} submanifold of some ℝ^{n}. Note that for any positive integer s, a locally definable C^{r} manifold is locally definably C^{s} imbeddable into some ℝ^{l} (1.3 [9]).

A locally definable C^{r} manifold (resp. An affine locally definable C^{r} manifold) G is a locally definable C^{r} group (resp. an affine locally definable C^{r} group) if G is a group and the group operations G × G → G, G → G are locally definable C^{r} maps.

P roof of T heorem 1.3. By the construction of the universal cover G̃ of G, G̃ is a C^{r} group whose charts are countable and π is a C^{r} map. Since G is a locally definable C^{r} group, every transition function is definable. □

References

[1] L. van den Dries, Tame topology and o-minimal structures, Lecture notes series 248, London Math. Soc. Cambridge Univ. Press (1998).

[2] L. van den Dries and C. Miller, Geometric categories and o-minimal structures, Duke Math. J. 84 (1996), 497-540.

[3] M.J. Edmundo, G.O. Jones and N.P. Peatfield, Invariance results for definable extension of groups, Arch. Math. Logic 50 (2011), 19–31.

[4] P. Heinzner, A.T. Huckleberry, F. Kutzschebauch, A real analytic version of Abels’ theorem and complexifications of proper Lie group actions. Complex analysis and geometry, Lecture Notes in Pure and Appl. Math., 173, Dekker, New York, (1996), 229–273.

[5] E. Hrushovski and A. Pillay, Groups definable in local fields and pseudo-finite fields, Israel J. Math. 85 (1994), 203–262.

[6] T. Kawakami, Definable C^{r} groups and proper definable actions, Bull. Fac. Ed. Wakayama Univ. Natur. Sci. 58 (2008), 9–18.

[7] T. Kawakami, Definable G CW complex structures of definable G sets and their applications, Bull. Fac. Ed. Wakayama Univ. Natur. Sci. 54 (2004), 1–15.

[8] T. Kawakami, Equivariant differential topology in an o-minimal expansion of the field of real numbers, Topology Appl. 123 (2002), 323–349.

[9] T. Kawakami, Locally definable C^{s}G manifold structures of locally definable C^{r}G manifolds, Bull. Fac. Ed. Wakayama Univ. Natur. Sci. 56 (2006), 1–12.

[10] T. Kawakami, Nash G manifold structures of compact or compactifiable C^{∞}G manifolds, J. Math. Soc. Japan 48 (1996), 321–331.

[11] A.L. Onishchik (Ed.), Lie groups and algebraic groups. Springer-Verlag, (1990).

[12] A. Pillay, On groups and fields definable in o-minimal structures, J. Pure Appl. Algebra 53 (1988), 239-255.

[13] M. Shiota, Geometry of subanalytic and semialgebraic sets, Progress in Mathematics 150, Birkhäuser, Boston, (1997).

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

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