Invariance results for definable extensions of groups

Mário J. Edmundo^{∗}

CMAF Universidade de Lisboa Av. Prof. Gama Pinto 2 1649-003 Lisboa, Portugal

Gareth O. Jones^{†}

School of Mathematics University of Manchester Oxford Road, Manchester, M13 9PL, UK

Nicholas J. Peatfield

5 Dunkerry Road Windmill Hill Bristol BS3 4LD, UK

August 9, 2010

Abstract

We show that in an o-minimal expansion of an ordered group finite definable extensions of a definable group which is defined in a reduct are already defined in the reduct. A similar result is proved for finite topological extensions of definable groups defined in o-minimal expansions of the ordered set of real numbers.

∗With partial support from the FCT (Fundação para a Ciência e Tecnologia) program POCI 2010 (Portugal/FEDER-EU) and FCT (Fundação para a Ciência e Tecnologia)

project PTDC/MAT/101740/2008.

†Supported by NSERC while a post-doc at McMaster University, Canada.

MSC: 03C64; 20E99. Keywords and phrases: O-minimal structures, definable groups.

1 Introduction

In this paper we work in an arbitrary o-minimal structure N and by definable we mean definable in N (possibly) with parameters. We will assume that the reader is familiar with the basic theory of o-minimality ([1]), definable groups (e.g. [2], [4], [8], [17], [18], [19], [20] and [21]) and of locally definable groups ([3], [5], [6] and [17]). When working in the category of locally definable groups we will assume that N is ℵ_{1}-saturated.

The main theorem of the paper is the following:

Theorem 1.1 Let K be an o-minimal structure definable in N and let G be a K-definably connected locally K-definable group defined without parameters and p_{H} : H → G a (locally) definable covering homomorphism. If there exists an admissible cover U = {U_{α}}_{α∈I} of G by open K-definable subsets defined without parameters over which p_{H} : H → G is trivial, then p_{H} : H → G is (locally) definably isomorphic to a (locally) K-definable covering homomorphism defined without parameters.

When N is an o-minimal expansion of an ordered group we have:

Corollary 1.2 Let K an o-minimal expansion of an ordered group definable in N and let G be a K-definably connected locally K-definable group defined without parameters and p_{H} : H → G a (locally) definable covering homomorphism. Then p_{H} : H → G is (locally) definably isomorphic to a (locally) K-definable covering homomorphism defined without parameters.

The next result is another consequence of the main theorem which was first observed in an unpublished manuscript by the authors and was quoted in the paper [13], where a different proof is presented, in connection with Pillay’s conjectures ([22]). The fundamental results of the theory definable groups, which includes real algebraic groups and semi-algebraic groups, hinted at a deep connection between definable groups and Lie groups, which was finally formulated in the paper [22] as Pillay’s conjecture for definable groups and the compact domination conjecture. Pillay’s conjecture is a nonstandard analogue of Hilbert’s 5^{o} problem for locally compact topological groups. On the other hand, the compact domination conjecture formalizes the idea that the quotient map in Pillay’s conjecture should be a kind of intrinsic “standard part map”. These conjectures lead to the development of new model theoretic tools as well as new geometric tools in o-minimality ([7], [8], [9], [10], [12], [13], [14], [15], [16]).

Corollary 1.3 Suppose that N is an o-minimal expansion of an ordered group. If H is a definably connected, semi-simple definable group, then there exists a definable real closed field R such that H is definably isomorphic in N, over parameters, to an R-semi-algebraic group H′ defined without parameters.

The technique used to prove the main theorem also gives with minor adaptations the following:

Theorem 1.4 Let K an ℵ_{1}-saturated elementary extension of an o-minimal expansion of the ordered set of real numbers. Let G be a K-definably connected locally K-definable group defined without parameters and p_{H} : H → G(ℝ) a topological covering homomorphism with countable discrete kernel. Then there are a locally K-definable group H′ and a locally K-definable covering homomorphism p_{H′} : H′ → G all defined without parameters such that H′(ℝ) = H and p_{H} : H → G(ℝ) is topologically isomorphic to p_{H′(ℝ)} : H′(ℝ) → G(ℝ). Moreover, if G is K-definable and the kernel of p_{H} : H → G(ℝ) is finite, then p_{H′} : H′ → G is K-definable defined without parameters.

2 On (locally) definable covering maps

Here we introduce some terminology and prove some preliminary results that will be useful later. A locally definable manifold (of dimension n) is a triple (S, (U_{i}, θ_{i})_{i≤κ}) with κ ≤ ω where:

• S = ⋃_{i≤κ} U_{i};

• each θ_{i} : U_{i} → N^{n} is an injection such that θ_{i}(U_{i}) is an open definable subset of N^{n};

• for all i, j, θ_{i}(U_{i} ∩ U_{j}) is an open definable subset of θ_{i}(U_{i}) and the transition maps θ_{ij} : θ_{i}(U_{i} ∩ U_{j}) → θ_{j}(U_{i} ∩ U_{j}) : x ↦ θ_{j}(θ_{i}^{−1}(x)) are definable homeomorphisms.

We call the (U_{i}, θ_{i})’s the definable charts of S. If κ < ω then S is a definable manifold.

A (locally) definable manifold S has a topology such that each U_{i} is open and the θ_{i}’s are homeomorphisms: a subset U of S is an open in the basis for this topology if and only if for each i, θ_{i}(U ∩ U_{i}) is an open definable subset of θ_{i}(U_{i}).

We say that a subset A of S is definable if and only if there is a finite I_{0} ⊆ κ such that A ⊆ ⋃_{i∈I_{0}} U_{i} and for each i ∈ I_{0}, θ_{i}(A ∩ U_{i}) is a definable subset of θ_{i}(U_{i}), and, a subset B of S is locally definable if and only if for each i, B ∩ U_{i} is a definable subset of S. We say that a (locally) definable manifold S is definably connected if it is not the disjoint union of two open and closed locally definable subsets.

Let U = {U_{α}}_{α∈I} be a cover of S by open definable subsets. We say that U is admissible if for each i ≤ κ, the cover {U_{α} ∩ U_{i}}_{α∈I} of U_{i} admits a finite subcover. If V = {V_{β}}_{β∈J} is another cover of S by open definable subsets, we say that V refines U, denoted V ≤ U, if there is a map ϵ : J → I such that V_{β} ⊆ U_{ϵ(β)} for all β ∈ J.

A map f : X ⟶ Y between locally definable manifolds with definable charts (U_{i}, θ_{i})_{i≤κ_{1}} and (V_{j}, δ_{j})_{j≤κ_{2}} respectively is a locally definable map if for every finite I ⊆ κ_{1} there is a finite J ⊆ κ_{2} such that:

• f(⋃_{i∈I} U_{i}) ⊆ ⋃_{j∈J} V_{j};

• the restriction f_{|} : ⋃_{i∈I} U_{i} ⟶ ⋃_{j∈J} V_{j} is a definable map between definable manifolds, i.e., for each i ∈ I and every j ∈ J with f(U_{i}) ∩ V_{j} ≠ ∅, δ_{j} ∘ f ∘ θ_{i}^{−1} : θ_{i}(U_{i}) ⟶ δ_{j}(V_{j}) is a definable map between definable sets.

Thus we have the category of (locally) definable manifolds with (locally) definable continuous maps.

Given definably connected locally definable manifold S, a locally definable manifold X and an admissible cover U = {U_{α}}_{α∈I} of S by open definable subsets, we say that a continuous surjective locally definable map p_{X} : X → S is a locally definable covering map trivial over U = {U_{α}}_{α∈I} if the following holds:

• p_{X}^{−1}(U_{α}) = ˙⋃_{i≤λ}U_{α}^{i} (a disjoint union) of open definable subsets of X with λ ≤ ω;

• each p_{X|U_{α}^{i}} : U_{α}^{i} → U_{α} is a definable homeomorphism,

If both S and X are definable manifolds we say that p_{X} : X → S is a definable covering map trivial over U = {U_{α}}_{α∈I}. A (locally) definable covering map p_{X} : X → G is a (locally) definable covering map trivial over some admissible cover U = {U_{α}}_{α∈I} of S by open definable subsets.

If Z is a countable discrete group, we say that a (locally) definable covering map p_{X} : X → S trivial over U = {U_{α}}_{α∈I} is a (locally) definable Z-covering map trivial over U = {U_{α}}_{α∈I} if in addition there exists a continuous (locally) definable action Z × X → X : (z, x) ↦ x^{z} of Z on X such that the following hold:

• For every x ∈ S we have an induced action of Z on p_{X}^{−1}(x) = {x_{i} : i ≤ |Z|} such that for any i, j there is a unique z ∈ Z such that x_{j} = x_{i}^{z}.

• For every α ∈ I we have an induced action of Z on p_{X}^{−1}(U_{α}) = {U_{α}^{i} : i ≤ |Z|} such that for any i, j there is a unique z ∈ Z such that U_{α}^{j} = (U_{α}^{i})^{z}.

(Note that when each U_{α} is definably connected, the first condition above implies the second). A (locally) definable Z-covering map p_{X} : X → G is a (locally) definable Z-covering map trivial over some admissible cover U = {U_{α}}_{α∈I} of S by open definable subsets.

We say that two (locally) definable Z-covering maps p_{X} : X → S and p_{Y} : Y → S (trivial over U) are (locally) definably homeomorphic if there is a (locally) definable homeomorphism F : X → Y such that:

• p_{X} = p_{Y} ∘ F.

• For every x ∈ X and z ∈ Z, we have F (x^{z}) = F (x)^{z}.

The first observation of the paper is:

Lemma 2.1 Let S be a definably connected locally definable manifold of dimension n, U = {U_{α}}_{α∈I} an admissible cover of S by open definably connected definable subsets which refines the collection of definable charts of S and p_{X} : X → S a surjective map.

(1) If for each α ∈ I we have

• p_{X}^{−1}(U_{α}) = ˙⋃_{i≤λ}U_{α}^{i} (a disjoint union) of subsets of X with λ ≤ ω;

• each p_{X|U_{α}^{i}} : U_{α}^{i} → U_{α} is a bijection;

then there exists a locally definable manifold structure on X of dimension n such that p_{X} : X → S is a locally definable covering map trivial over U = {U_{α}}_{α∈I}.

(2) If Z is a countable discrete group and in addition there is an action Z × X → X : (z, x) ↦ x^{z} of Z on X such that, for each α ∈ I the following two conditions hold

• for every x ∈ S we have an induced action of Z on p_{X}^{−1}(x) = {x_{i} : i ≤ |Z|} such that for any i, j there is a unique z ∈ Z such that x_{j} = x_{i}^{z}; and

• there is an induced action of Z on p_{X}^{−1}(U_{α}) = {U_{α}^{i} : i ≤ |Z|} such that for any i, j there is a unique z ∈ Z such that U_{α}^{j} = (U_{α}^{i})^{z};

then p_{X} : X → S is a locally definable Z-covering map trivial over U = {U_{α}}_{α∈I}.

Proof. Let (U_{i}, θ_{i})_{i≤κ} be the definable charts of S and let ι : I → κ be a map such that U_{α} ⊆ U_{ι(α)} for all α ∈ I. Let θ_{α} := θ_{ι(α)|U_{α}}. Since U = {U_{α}}_{α∈I} is an admissible cover of S by open definable subsets U has a countable subcover. Thus, without loss of generality, we may assume that I is countable.

For each α ∈ I and i ≤ λ, set δ_{α,i} : U_{α}^{i} → θ_{α}(U_{α}) : x ↦ θ_{α} ∘ p_{X|U_{α}^{i}}. We claim that the (U_{α}^{i}, δ_{α,i})’s are the charts making X into a locally definable manifold of dimension n. Clearly we have:

• X = ∪{U_{α}^{i} : α ∈ I, i ≤ λ};

• each δ_{α,i} : U_{α}^{i} → N^{n} is an injection such that δ_{α,i}(U_{α}^{i}) = θ_{α}(U_{α}) is an open definable subset of N^{n}; • for all α, i, β, j, δ_{α,i}(U_{α}^{i}∩U_{β}^{j}) = θ_{α}(U_{α}∩U_{β}) is an open definable subset

of δ_{α,i}(U_{α}^{i}) = θ_{α}(U_{α}) and the transition maps δ_{(α,i),(β,j)} : δ_{α,i}(U_{α}^{i} ∩ U_{β}^{j}) → δ_{β,j}(U_{α}^{i} ∩ U_{β}^{j}) : x ↦ δ_{β,j}(λ_{α,i}^{−1}(x)) = θ_{β}(θ_{α}^{−1}(x)) are definable homeomorphisms.

When X is equipped with this locally definable manifold structure, the map p_{X} : X → S is a continuous locally definable map since θ_{α} ∘ p_{X|U_{α}^{i}} ∘ δ_{α,i}^{−1} : δ_{α,i}(U_{α}^{i}) → θ_{α}(U_{α}) is the identity map on θ_{α}(U_{α}). Furthermore, by the hypothesis on p_{X} : X → S, this map is a locally definable covering map trivial over U = {U_{α}}_{α∈I}.

(2) Since Z is a countable discrete group, to show that the action Z × X → X : (z, x) ↦ x^{z} of Z on X is continuous and locally definable it is enough to show that for each z ∈ Z, the map X → X : x ↦ x^{z} is continuous and locally definable. But through the charts, this map is δ_{α,i}(U_{α}^{i}) → δ_{α,j}(U_{α}^{j}) : x ↦ δ_{α,j}((δ_{α,i}^{−1}(x))^{z}), where j is the unique element such that U_{α}^{j} = (U_{α}^{i})^{z}, and therefore it is the identity map on θ_{α}(U_{α}). Thus p_{X} : X → S is a locally definable Z-covering map trivial over U = {U_{α}}_{α∈I}. □

An immediate consequence of Lemma 2.1 is:

Corollary 2.2 Let K be an o-minimal structure definable in N. Let S be a K-definably connected locally K-definable manifold defined without parameters and p_{X} : X → S a (locally) definable covering (resp. Z-covering) map. Suppose that there exists an admissible cover V = {V_{α}}_{α∈I} of S by K-definable open subsets defined without parameters such that:

• p_{X} : X → S is trivial over V = {V_{α}}_{α∈I}.

• V = {V_{α}}_{α∈I} refines the collection of definable charts of S.

Then p_{X} : X → S is (locally) definably homeomorphic to a (locally) Kdefinable covering (resp. Z-covering) map trivial over V = {V_{α}}_{α∈I}.

Similarly we have:

Corollary 2.3 Let K be an o-minimal expansion of the ordered set of real numbers. Let S be a K-definably connected locally K-definable manifold defined without parameters and p_{X} : X → S a topological covering (resp. Z-covering) map with countable fibers. Suppose that there exists an admissible cover V = {V_{α}}_{α∈I} of S by K-definable open subsets defined without parameters such that:

• p_{X} : X → S is trivial over V = {V_{α}}_{α∈I}.

• V = {V_{α}}_{α∈I} refines the collection of definable charts of S.

Then p_{X} : X → S is topologically homeomorphic to a locally K-definable covering (resp. Z-covering) map trivial over V = {V_{α}}_{α∈I}. Moreover, if S is K-definable and the fibers of p_{X} : X → S are finite, then p_{X} : X → S is topologically homeomorphic to a K-definable covering (resp. Z-covering) map trivial over V = {V_{α}}_{α∈I}

The following lemma will also be quite useful later. In recent work by Edmundo, Eleftheriou and Prelli the assumptions of this lemma are proved. Note however, that in this paper we apply the lemma for locally definable groups in which case the assumptions were verified in [6].

Lemma 2.4 Let K be an o-minimal expansion of an ordered group (resp. of the ordered set of real numbers). Let S be a K-definably connected locally K-definable manifold and suppose that there exists an admissible cover U = {U_{α}}_{α∈I} of S by K-definable open subsets of S such that each U_{α} is K-definably connected and K-definably simply connected (resp. connected and simply connected). Then every locally K-definable covering map (resp. topological covering map) p_{X} : X → S is trivial over U = {U_{α}}_{α∈I}.

Proof. Let p_{X} : X → S be a locally K-definable covering map (resp. a topological covering map). For each α ∈ I set p_{|X}^{−1}(U_{α}) = ˙⋃_{i≤λ_{α}}U_{α}^{i} with λ_{α} ≤ ω where the U_{α}^{i}’s are the K-definably connected components (resp. connected components) of p_{|X}^{−1}(U_{α}). We have to show that:

• each p_{|X} : U_{α}^{i} → U_{α} is a K-definable (resp. topological) homeomorphism;

• λ_{α} = λ_{β} for all α, β ∈ I.

Note that the first point implies that λ_{α} = |p_{X}^{−1}(x)| for all x ∈ U_{α}. In particular, if U_{α} ∩ U_{β} ≠ ∅ then λ_{α} = λ_{β}. Since S is K-definably connected, we must have also λ_{α} = λ_{β} for all α, β ∈ I as required. On the other hand, since p_{X} : X → S is a locally K-definable covering map (resp. a topological covering map) it is by definition a local homeomorphism. So it is enough to show that each p_{X|U_{α}^{i}} : U_{α}^{i} → U_{α} is a bijection.

We show first that p_{X|U_{α}^{i}} : U_{α}^{i} → U_{α} is surjective. Let u ∈ U_{α}^{i} and x = p_{X|U_{α}^{i}}(u) ∈ U_{α}. Let y ∈ U_{α} and since U_{α} is K-definably path connected (resp. path connected), let γ be a K-definable path (resp. path) in U_{α} from x to y. By [6] Lemma 2.7 (1) (resp. [11] Chapter 11, Proposition 11.6), there is a unique K-definable path (resp. path) γ̃ in X, lifting γ, and starting at u. The image of γ̃ is a K-definably connected (resp. connected) subset of ˙⋃_{i≤λ_{α}}U_{α}^{i} and intersects U_{α}^{i}. Thus this image is contained in U_{α}^{i} and the endpoint v of γ̃ is also contained in U_{α}^{i} and is mapped by p_{X|U_{α}^{i}} to y.

Suppose that u, v ∈ U_{α}^{i} are such that p_{X|U_{α}^{i}}(u) = x = p_{X|U_{α}^{i}}(v). Since U_{α}^{i} is K-definably connected (resp. connected), it is K-definably path connected by [1] Chapter 6, Proposition 3.2 (resp. path connected because it also locally path connected as p_{X} is a local homeomorphism and U_{α} is locally path connected). Let γ be a K-definable path (resp. path) in U_{α}^{i} from u to v. Then σ = p_{X|U_{α}^{i}} ∘ γ is a K-definable loop (resp. loop) in U_{α} at x and γ is the unique K-definable lifting (resp. lifting) of σ to U_{α}^{i} starting at u. Since the o-minimal fundamental groups (resp. the topological fundamental groups) π_{1}(U_{α}) are trivial, it follows from [6] Remark 2.9 (1) (resp. [11] Chapter 13, Exercise 13.2) that u = v. So p_{X|U_{α}^{i}} : U_{α}^{i} → U_{α} is injective.

□

3 On definable covering homomorphisms

Recall that a group (G, ·) is a locally definable group over A, with A ⊆ N and |A| < ℵ_{1}, if there is a countable collection {Z_{i} : i ∈ I} of definable subsets of N^{n}, all definable over A, such that:

• G = ∪{Z_{i} : i ∈ I};

• for every i, j ∈ I there is k ∈ I such that Z_{i} ∪ Z_{j} ⊆ Z_{k};

• the restriction of the group multiplication to Z_{i} × Z_{j} is a definable map over A into N^{n}.

Given two locally definable groups H and G over A, we say that H is a locally definable subgroup of G over A if H is a subgroup of G. A homomorphism α : G ⟶ H between locally definable groups over A is called a locally definable homomorphism over A if for every definable subset Z ⊆ G defined over A, the restriction α_{|Z} is a definable map over A.

In the terminology of [17], locally definable groups (respectively homomorphisms) are ⋁-definable groups (respectively homomorphisms). Therefore, every locally definable group G ⊆ N^{n} over A is equipped with a unique topology τ, called the τ-topology, such that: (i) (G, τ) is a topological group; (ii) every generic element of G has an open definable neighborhood U ⊆ N^{n} such that U ∩ G is τ-open and the topology which U ∩ G inherits from τ agrees with the topology it inherits from N^{n}; (iii) locally definable homomorphisms between locally definable groups are continuous with respect to the τ topologies. Note also that when G is a definable group, then its τtopology coincides with the its t-topology from [21]. In fact, just like in the definable case ([21]) we have the following (see the proof of Proposition 3.3 in [6]):

Proposition 3.1 ([6]) Let G be a locally definable group of dimension k and defined without parameters. Then there is a countable collection {O_{s} : s ∈ S} of τ-open definably τ-connected definable subsets of G defined without parameters and a countable collection {θ_{s} : O_{s} → N^{k} : s ∈ S} of definable homeomorphisms defined without parameters such that:

• G = ∪{O_{s} : s ∈ S};

• each θ_{s} : O_{s} → N^{k} is a definable homeomorphism onto an open cell in N^{k};

• for all s, t, θ_{s}(O_{s} ∩ O_{t}) is an open definable subset of θ_{s}(O_{s}) and the transition maps θ_{st} : θ_{s}(O_{s} ∩ O_{t}) → θ_{t}(O_{s} ∩ O_{t}) : x ↦ θ_{t}(θ_{s}^{−1}(x)) are definable homeomorphisms.

In particular, (O_{s}, θ_{s})_{s∈S} is a locally definable manifold structure on G.

We say that a locally definable group G is definably connected if it so as a locally definable manifold. This is equivalent to notion of definably τ-connected from [5] and it corrigendum.

Another result that will be useful below is the following (see the proof of Theorem 3.6 in [3]):

Theorem 3.2 ([3]) Let p_{H} : H ⟶ G be a surjective locally definable homomophism over A between locally definable groups over A. If Kerp_{H} has dimension zero, then p_{H} : H ⟶ G is a locally definable covering homomorphism over A. In fact, p_{H} : H ⟶ G is a locally definable Kerp_{H}-covering homomorphism over A.

So we often call a surjective (locally) definable homomorphism p_{H} : H → G with a (necessarily (locally) definable) kernel Z of dimension zero a (locally) definable Z-covering homomorphism (trivial over some admissible cover U = {U_{α}}_{α∈I} of G by open definable subsets). We say that two (locally) definable Z-covering homomorphisms p_{H} : H → G and p_{H′} : H′ → G (trivial over some admissible cover U = {U_{α}}_{α∈I} of G by open definable subsets) are (locally) definably isomorphic if there is a (locally) definable isomorphism F : H → H′ such that:

• p_{H} = p_{H′} ∘ F.

• For every x ∈ X and z ∈ Z, we have F (x^{z}) = F (x)^{z}.

We are now ready to present the proof of Theorem 1.1. The argument of this proof applies with obvious modifications also to the proof of the following Theorem which is needed in the proof of Theorem 1.4. Thus we will run the two proofs in parallel.

Theorem 3.3 Let K be an ℵ_{1}-saturated elementary extension of an o-minimal expansion of the ordered set of real numbers. Let G be a K-definably connected locally K-definable group defined without parameters and p_{H} : H → G(ℝ) a topological covering homomorphism with countable discrete kernel. Suppose that there exists an admissible cover U = {U_{α}}_{α∈I} of G by open K-definable subsets defined without parameters such that p_{H} : H → G(ℝ) is trivial over U(ℝ) = {U_{α}(ℝ)}_{α∈I}. Then there are a locally K-definable group H′ and a locally K-definable covering homomorphism p_{H′} : H′ → G all defined without parameters such that H′(ℝ) = H and p_{H} : H → G(ℝ) is topologically isomorphic to p_{H′(ℝ)} : H′(ℝ) → G(ℝ). Moreover, if G is K-definable and the kernel of p_{H} : H → G(ℝ) is finite, then p_{H′} : H′ → G is K-definable defined without parameters.

Proof of Theorem 1.1 (resp. Theorem 3.3): Let K, G and p_{H} : H → G (resp. p_{H} : H → G(ℝ)) be as in the assumptions of Theorem 1.1 (resp. Theorem 3.3). Assume also that there exists an admissible cover U = {U_{α}}_{α∈I} of G by open K-definable subsets defined without parameters such that p_{H} : H → G (resp. p_{H} : H → G(ℝ)) is trivial over U = {U_{α}}_{α∈I} (resp. U(ℝ) = {U_{α}(ℝ)}_{α∈I}). Let K(ℝ) is the o-minimal expansion of the ordered set of real number of which K of Theorem 3.3 is an ℵ_{1}-saturated elementary extension.

By taking a refinement we may assume without loss of generality that the admissible cover U = {U_{α}}_{α∈I} of G refines the collection of definable charts of G. By Corollary 2.2 (resp. Corollary 2.3) we may assume that p_{H} : H → G (resp. p_{H} : H → G(ℝ)) is a locally K-definable covering map defined without parameters trivial over U = {U_{α}}_{α∈I} (resp. U(ℝ) = {U_{α}(ℝ)}_{α∈I}). Thus it remains to show that the group operation on H is locally K-definable and defined without parameters.

To simplify the notation below we will still write K for K(ℝ), G for G(ℝ) and U = {U_{α}}_{α∈I} for U(ℝ) = {U_{α}(ℝ)}_{α∈I}.

We have the following commutative diagram

\xymatrix{H × H \ar[d]_{p_{H}×p_{H}} \ar[r]^{μ_{H}} & H \ar[d]^{p_{H}} \\ G × G \ar[r]^{μ_{G}} & G}

where μ_{H} (resp. μ_{G}) is the group operation in H (resp. in G). For α ∈ I, let p_{H}^{−1}(U_{α}) = ˙⋃_{i≤λ}U_{α}^{i}. Since the U_{α}^{i}’s are K-definable and defined without parameters, to reach our goal it is enough to show that each restriction μ_{H|U_{α}^{i}×U_{β}^{j}} is K-definable and defined without parameters.

Fix α, β ∈ I and i, j ≤ λ. For γ ∈ I and k ≤ λ consider the subsets

A_{α,β;γ}^{i,j;k} = {(x, y) ∈ U_{α}^{i} × U_{β}^{j}|μ_{H}(x, y) ∈ U_{γ}^{k}}

and

A_{α,β;γ}^{i,j} = {(x, y) ∈ U_{α}^{i} × U_{β}^{j}|μ_{H}(x, y) ∈ p_{H}^{−1}(U_{γ})}

of U_{α}^{i} × U_{β}^{j}. Consider also the K-definable subset

A_{α,β;γ} = {(u, v) ∈ U_{α} × U_{β}|μ_{G}(u, v) ∈ U_{γ}}

of U_{α} × U_{β} defined without parameters. By the commutativity of the previous diagram we clearly we have

(p_{H|U_{α}^{i}})^{−1} × (p_{H|U_{β}^{j}})^{−1}(A_{α,β;γ}) = A_{α,β;γ}^{i,j}

and so A_{α,β;γ}^{i,j} is K-definable and defined without parameters. Clearly

A_{α,β;γ}^{i,j} = ˙⋃_{k≤λ}A_{α,β;γ}^{i,j;k}

and by the continuity of μ_{H|A_{α,β;γ}^{i,j}} : A_{α,β;γ}^{i,j} → p_{H}^{−1}(U_{γ}), each A_{α,β;γ}^{i,j;k} is clopen

in A_{α,β;γ}^{i,j}.

Claim 3.4 Each A_{α,β;γ}^{i,j;k} is K-definable and defined without parameters and

A_{α,β;γ}^{i,j} = A_{α,β;γ}^{i,j;k_{1}} ˙∪ ⋯ ˙∪ A_{α,β;γ}^{i,j;k_{γ}}

for some finite k_{γ}.

Proof. Take a cell decomposition C of A_{α,β;γ}^{i,j} in the prime model of K over the empty set. Since each cell is N -definably connected (resp. is connected) and each A_{α,β;γ}^{i,j;k} which is non empty is N -definable and clopen (resp. clopen), each cell in C is contained in an N -definably connected component (resp. connected component) of some A_{α,β;γ}^{i,j;k}. Therefore, each non empty A_{α,β;γ}^{i,j;k} is a finite union of cells in C and therefore it is K-definable and defined without parameters. On the other hand, since A_{α,β;γ}^{i,j} has finitely many K-definably connected components and each A_{α,β;γ}^{i,j;k} which is non empty is clopen and

K-definable we must have A_{α,β;γ}^{i,j} = A_{α,β;γ}^{i,j;k_{1}} ˙∪ ⋯ ˙∪ A_{α,β;γ}^{i,j;k_{γ}} for some finite k_{γ}. □

Since for each k ∈ {k_{1}, …, k_{γ}} we have

μ_{H|A_{α,β;γ}^{i,j;k}} = (p_{H|U_{γ}^{k}})^{−1} ∘ μ_{G} ∘ (p_{H|U_{α}^{i}} × p_{H|U_{β}^{j}})

it follows from Claim 3.4 that each such μ_{H|A_{α,β;γ}^{i,j;k}} is K-definable and defined without parameters.

Claim 3.5 There is a finite subset J_{α,β}^{i,j} ⊆ I such that

U_{α}^{i} × U_{β}^{j} = ⋃{A_{α,β;γ}^{i,j;k}|γ ∈ J_{α,β}^{i,j}, k ∈ {k_{1}, …, k_{γ}}}.

Proof. Since μ_{G}(U_{α} × U_{β}) is a K-definable subset of G defined without parameters and U = {U_{α}}_{α∈I} is an admissible cover of G by open K-definable subsets defined without parameters, there is a finite subset J_{α,β}^{i,j} ⊆ I such that μ_{G}(U_{α} × U_{β}) ⊆ ⋃{U_{γ}|γ ∈ J_{α,β}^{i,j}}. Since p_{H} ∘ μ_{H} = μ_{G} ∘ (p_{H} × p_{H}) we have μ_{H}(U_{α}^{i} × U_{β}^{j}) ⊆ ⋃{p_{H}^{−1}(U_{γ})|γ ∈ J_{α,β}^{i,j}} and the result follows from this by the definition of the A_{α,β;γ}^{i,j;k}’s. □

It follows from this Claim 3.5 and the fact that each μ_{H|A_{α,β;γ}^{i,j;k}} with k ∈ {k_{1}, …, k_{γ}} is K-definable and defined without parameters that each μ_{H|U_{α}^{i}×U_{β}^{j}} is K-definable and defined without parameters as required concluding in this way the proof of Theorem 1.1 (resp. Theorem 3.3). Moreover, if p_{H} : H → G (resp. p_{H} : H → G(ℝ) is definable (resp. has finite kernel and G is K-definable), then p_{H} : H → G is definably isomorphic to a K-definable covering homomorphism defined without parameters (resp. p_{H′} : H′ → G is K-definable defined without parameters). □

For the proof of Corollary 1.2 we require the following lemma:

Lemma 3.6 Let K be an o-minimal expansion of an ordered group and G a K-definably connected locally K-definable group defined without parameters. Then there exists an admissible cover U = {U_{α}}_{α∈I} of G by K-definable open subsets of G defined without parameters such that each U_{α} is K-definably connected and K-definably simply connected. In particular, by Lemma 2.4, every locally K-definable covering map p_{X} : X → G is trivial over U = {U_{α}}_{α∈I}.

On the other hand for the proof of Theorem 1.4 we require the following analogue of Lemma 3.6:

Lemma 3.7 Let K be an ℵ_{1}-saturated elementary extension of an o-minimal expansion of the ordered set of real numbers. Let G a K-definably connected locally K-definable group defined without parameters. Then there exists an admissible cover U = {U_{α}}_{α∈I} of G by K-definable open subsets of G defined without parameters such that each U_{α}(ℝ) is connected and simply connected. In particular, by Lemma 2.4, every topological covering map p_{X} : X → G(ℝ) is trivial over U(ℝ) = {U_{α}(ℝ)}_{α∈I}.

As before the proofs of Lemma 3.6 and Lemma 3.7 will run in parallel.

Proof of Lemma 3.6 (resp. Lemma 3.7): Let K and G be as in the assumptions of Lemma 3.6 (resp. Lemma 3.7). Let K(ℝ) is the o-minimal expansion of the ordered set of real number of which K of Lemma 3.7 is an ℵ_{1}-saturated elementary extension.

By Theorem 3.1, let U = {U_{α}}_{α∈I} be an admissible cover of G by Kdefinably connected, open K-definable subsets of G defined without parameters each of which is K-definably homeomorphic over the empty set to an open K-cell in K^{n} where n = dimG. By [1] Chapter 6, Proposition 3.2, the U_{α}’s are K-definably connected, K-definably path connected and by [6] Proposition 3.3, the o-minimal fundamental groups π_{1}(U_{α}) are trivial (resp. by similar arguments, the U_{α}(ℝ)’s are connected, path connected and the topological fundamental groups π_{1}(U_{α}(ℝ)) are trivial).

□

Proof of Corollary 1.2: Let K an o-minimal expansion of an ordered group definable in N and let G be a K-definably connected locally K-definable group defined without parameters and p_{H} : H → G a locally definable covering homomorphism.

By definition p_{H} : H → G is a locally definable covering map trivial over some admissible cover W = {W_{α}}_{α∈L} of G by definable in N open subsets of G. Let J be the o-minimal expansion of K expanded by all the definable in N subsets of each K^{m}. In particular, the W_{α}’s are J - definable open subsets of G defined without parameters. By Theorem 1.1 applied to the definable o-minimal structure J in N, p_{H} : H → G is locally definably isomorphic to a locally J -definable covering homomorphism. By Lemmas 2.4 and 3.6 and Theorem 1.1 applied to K and J instead of K and N, p_{H} : H → G is locally J -definably isomorphic to a locally Kdefinable covering homomorphism defined without parameters. Moreover, if p_{H} : H → G is definable, then p_{H} : H → G is definably isomorphic to a K-definable covering homomorphism defined without parameters. □

Proof of Corollary 1.3: Let G be a definably connected, semi-simple definable group, i.e. it is non abelian and has no infinite normal abelian subgroups. In this case, the center Z(G) of G is finite and by [18] and [19] (the proof of Theorem 5.1 there), there is a definable real closed field R and there are R-definably simple, R-definable groups G_{i} defined without parameters such that G/Z(G) is definably isomorphic to G_{1} ×⋯×G_{l}. Now apply Corollary 1.2 to the definable covering homomorphism h : G ⟶ G_{1} × ⋯ × G_{l} which is the composition of the quotient definable homomorphism G ⟶ G/Z(G) and the definable isomorphism G/Z(G) ≃ G_{1} × ⋯ × G_{l}. □

Proof of Theorem 1.4: Let K an ℵ_{1}-saturated elementary extension of an o-minimal expansion of the ordered set of real numbers. Let G be a Kdefinably connected locally K-definable group defined without parameters and p_{H} : H → G(ℝ) a topological covering homomorphism with countable discrete kernel.

By Lemmas 2.4 and 3.7 and Theorem 3.3 there is a locally K-definable covering homomorphism p_{H′} : H′ → G defined without parameters such that p_{H} : H → G(ℝ) is topologically isomorphic to p_{H′(ℝ)} : H′(ℝ) → G(ℝ). Moreover, if G is K-definable and the kernel of p_{H} : H → G(ℝ) is finite, then p_{H′} : H′ → G is K-definable defined without parameters.

□

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