Definable proper actions and equivariant definable Tietze extension

Tomohiro Kawakami

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

Abstract

Let N = (R, +, ·, <, …) be an o-minimal expansion of the standard structure of a real closed field R. Let G be a definable group and X a definable proper definable G set. We prove that X has only finitely many orbit types. We also prove equivariant definable Tietze extension theorem.

2010 Mathematics Subject Classification. 14P10, 57S10, 03C64. Keywords and P hrases. Finiteness of orbit types, equivariant definable Tietze extension theorem, o-minimal, real closed fields.

1 . Introduction.

Let N = (R, +, ·, <, …) be an o-minimal expansion of the standard structure of a real closed field R.

General references on o-minimal structures are [2], [3], and also see [9]. Any definable category is a generalization of the semialgebraic category and the definable category on R = (R, +, ·, <) coincides with the semialgebraic one. It is known in [8] that there exist uncountably many o-minimal expansions of the field ℝ of real numbers.

Let G be a definable group. A definable G set means a pair consisting of a definable set X and a group action ϕ : G × X → X such that ϕ is definable. A definable map between definable sets is called definably proper if the inverse image of every definably compact definable set is definably compact. We call a definable G set X a proper definable G set if the map G × X → X × X defined by (g, x) ↦ (gx, x) is definably proper.

Let G be a definable group. We can define orbit types as well as G is definably compact ([5]). Theorem 1.1. Let G be a definable group. Then every proper definable G set X has only finitely many orbit types.

Theorem 1.1 is proved the case where R is the field ℝ of real numbers ([5]).

The following theorem is an equivariant version of definable Tietze extension theorem [1]

Theorem 1.2. Let G be a definably compact definable group, X a definable G set and A a G invariant definably compact definable subset of X. Every G invariant definable function f : A → R is extensible to a G invariant definable function F : X → R with F |A = f.

2 . 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. 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 definable subset X of R^{n} is definably compact if for every definable map f : (a, b)_{R} → X, there exist the limits lim_{x→a+0} f(x), lim_{x→b−0} f(x) in X, where (a, b)_{R} = {x ∈ R|a ≤ x < b}, −∞ ≤ a < b ≤ ∞. A definable subset X of R^{n} is definably compact if and only if X is closed and bounded ([7]). Note that if X is a definably compact definable set and f : X → Y is a definable map, then f(X) is definably compact.

We say that two homogeneous proper definable G sets are equivalent if they are definably G homeomorphic. Let (G/H) denote the equivalence class of G/H. The set of all equivalence classes of homogeneous proper definable G sets has a natural order defined as (X) ≥ (Y) if there exists a definable G map X → Y. By the definition the reflexivity and the transitivity clearly hold If (X) = (G/H) and (Y) = (G/K), then (X) ≥ (Y) if and only if H is conjugate to a definable subgroup of K. By a way similar to the proof of 4.1 [5], we have the following lemma.

Lemma 2.1. Let G be a definable group, H a definable subgroup of G and g ∈ G. If gHg^{−1} ⊂ H, then gHg^{−1} = H.

By Lemma 2.1, the anti-symmetry is true. By a way to similar to the proof of 1.1

[5], we have Theorem 1.1.

Note that every definable subgroup of a definable group is closed ([6]) and a closed subgroup of a definable group is not necessarily definable. For example ℤ is a closed subgroup of ℝ but not a definable subgroup of ℝ.

Recall existence of definable quotient.

Theorem 2.2 (Existence of definable quotient, 10.2.18 [2]). Let G be a

definably compact definable group and X a definable G set. Then the orbit space X/G exists as a definable set, and the orbit map π : X → X/G is definable, surjective and definably proper.

The following theorem is the topological case of Tietze extension theorem.

Theorem 2.3 (Tietze extension the-

orem). Let X be a normal space and A a closed subset of X. Then every continuous map f : A → ℝ is extensible to a continuous map F : X → ℝ with F |A = f.

The following theorem is the definable case of Tietze extension theorem.

Theorem 2.4 (Definable Tietze extension theorem, [1]). Let A be a defin-

able closed subset of R^{n}. Then every definable map f : A → R is extensible to a definable map F : R^{n} → R with F |A = f.

A definable map f : X → Y is definably closed if for any definable closed subset A of X, f(A) is a definable closed subset of Y.

Theorem 2.5 ([4]). Let f : X → Y be

a definable map. Then f is definably proper if and only if f is definably closed and has definably compact fibers.

P roof of T heorem 1.2. By Theorem 2.1, X/G exists as a definable set in R^{n} and the projection π : X → X/G is a surjective definable definably proper map. By Theorem 2.4 and A is definably compact, π(A) is a definable closed subset of R^{n}. Since f is a G invariant definable map, it induces a definable map f′ : f(A) → R with f = π ∘ f′. By Theorem 2.2, there exists a definable map F : R^{n} → R with F |f(A) = f′. Hence H = π ∘ F is the required map.

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