The universal covering homomorphism in o-minimal expansions of groups

Mário J. Edmundo^{∗}

CMAF Universidade de Lisboa Av. Prof. Gama Pinto 2 1649-003 Lisboa, Portugal edmundo@cii.fc.ul.pt

Pantelis E. Eleftheriou^{†}

Department of Mathematics University of Notre Dame Notre Dame, IN 46556, USA pelefthe@nd.edu

February 19, 2006

Abstract

Suppose that G is a definably connected, definable group in an ominimal expansion of an ordered group. We show that the o-minimal universal covering homomorphism p̃ : G̃ ⟶ G is a locally definable covering homomorphism and π_{1}(G) is isomorphic to the o-minimal fundamental group π(G) of G defined using locally definable covering homomorphisms.

∗With partial support from the FCT (Fundação para a Ciência e Tecnologia), program POCTI (Portugal/FEDER-EU).

†Partially supported by NSF grant DMS-02-45167 (Cholak). MSC: 03C64, 20E99.

Keywords and phrases: O-minimal structures, universal covers, definable groups. Revised:

June 1, 2007.

1 Introduction

Let R be an o-minimal expansion of an ordered group (R, 0, +, <). The structure R will be fixed throughout and will be assumed to be ℵ_{1}-saturated. By definable we will mean definable in R with parameters.

In the paper [3] the first author introduced a notion of o-minimal fundamental group and o-minimal universal covering homomorphism for definable groups (or more generally for locally definable groups) in arbitrary o-minimal structures which we now recall.

First recall that a group (G, ·) is a locally definable group over A, with A ⊆ R and |A| < ℵ_{1}, if there is a countable collection {Z_{i} : i ∈ I} of definable subsets of R^{n}, all definable over A, such that: (i) G = ∪{Z_{i} : i ∈ I}; (ii) for every i, j ∈ I there is k ∈ I such that Z_{i} ∪ Z_{j} ⊆ Z_{k} and (iii) the restriction of the group multiplication to Z_{i} × Z_{j} is a definable map over A into R^{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 [9], locally definable groups (respectively homomorphisms) are ⋁-definable groups (respectively homomorphisms). Therefore, every locally definable group G ⊆ R^{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 ⊆ R^{n} such that U ∩ G is τ-open and the topology which U ∩ G inherits from τ agrees with the topology it inherits from R^{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 τ-topology coincides with the its t-topology from [10].

Definition 1.1 A locally definable homomorphism p : H ⟶ G over A between locally definable groups over A is called a locally definable covering homomorphism if p is surjective and there is a family {U_{l} : l ∈ L} of τ-open definable subsets of G over A such that G = ∪{U_{l} : l ∈ L} and, for each l ∈ L, p^{−1}(U_{l}) is a disjoint union of τ-open definable subsets of H over A, each of which is mapped homeomorphically by p onto U_{l}.

We call {U_{l} : l ∈ L} a p-admissible family of definable τ-neighborhoods over A.

We denote by Cov(G) the category whose objects are locally definable covering homomorphisms p : H ⟶ G (over some A with |A| < ℵ_{1}) and whose morphisms are surjective locally definable homomorphisms r : H ⟶ K (over some A with |A| < ℵ_{1}) such that q ∘ r = p, where q : K ⟶ G is a locally definable covering homomorphism (over some A with |A| < ℵ_{1}). Let p : H ⟶ G and q : K ⟶ G be locally definable covering homomorphisms. If r : H ⟶ K is a morphism in Cov(G), then by [3] Theorem 3.6, r : H ⟶ K is a locally definable covering homomorphism.

Definition 1.2 The category Cov(G) and its full subcategory Cov^{0}(G) with objects h : H ⟶ G such that H is a definably connected locally definable group, form inverse systems ([3] Corollary 3.7 and Lemma 3.8). The inverse limit p̃ : G̃ ⟶ G of the inverse system Cov^{0}(G) is called the (o-minimal) universal covering homomorphism of G.

The kernel of the universal covering homomorphism p̃: G̃ ⟶ G of G is called the (o-minimal) fundamental group of G and is denoted by π(G).

Inverse limits of inverse systems of groups always exist in the category of groups ([11] Proposition 1.1.1), but in general we do not know if the ominimal universal covering homomorphism p̃: G̃ ⟶ G is locally definable. The main result of this paper is that this is the case in o-minimal expansions of groups.

On the other hand, in the paper [5], the second author and S. Starchenko use definable t-continuous paths to define the o-minimal fundamental group π_{1}(G) of a definably t-connected, definable group G following the classical case in [7] and the case in o-minimal expansions of fields treated by Berarducci and Otero in [1]. We will adapt that definition to the category of locally definable groups. As in [5] we will run the definition in parallel with respect to the τ-topology of a definably connected locally definable group G and the usual topology on an arbitrary definable subset X of R^{n}.

A (τ-)path α : [0, p] ⟶ X (α : [0, p] ⟶ G) is a (τ-)continuous definable map. A (τ-)path α : [0, p] ⟶ X (α : [0, p] ⟶ G) is a (τ-)loop if α(0) = α(p). A concatenation of two (τ-)paths γ : [0, p] ⟶ X (γ : [0, p] ⟶ G) and δ : [0, q] ⟶ X (δ : [0, q] ⟶ G) with γ(p) = δ(0) is a (τ-)path γ · δ : [0, p + q] ⟶ X (γ · δ : [0, p + q] ⟶ G) with:

(γ · δ)(t) = \{ \begin{matrix}γ(t) & if t ∈ [0, p] \\ δ(t − p) & if t ∈ [p, p + q]. \end{matrix}

Given two definable (τ-)continuous maps f, g : Y ⊆ R^{m} ⟶ X (f, g : Y ⊆ R^{m} ⟶ G), we say that a definable (τ-)continuous map F (t, s) : Y ×[0, q] ⟶ X (F (t, s) : Y × [0, q] ⟶ G), is a (τ-)homotopy between f and g if f = F_{0} and g = F_{q}, where ∀s ∈ [0, q], F_{s} := F (·, s). In this situation we say that f and g are (τ-)homotopic, denoted f ∼ g (f ∼_{τ} g).

Definition 1.3 Two (τ-)paths γ : [0, p] ⟶ X (γ : [0, p] ⟶ G), δ : [0, q] ⟶ X (δ : [0, q] ⟶ G), with γ(0) = δ(0) and γ(p) = δ(q), are called (τ-)homotopic if there is some t_{0} ∈ [0, min{p, q}], and a (τ-)homotopy F (t, s) : [0, max{p, q}] × [0, r] ⟶ X (F (t, s) : [0, max{p, q}] × [0, r] ⟶ G), for some r > 0 in R, between

γ_{|[0,t_{0}]} · c · γ_{|[t_{0},p]} and δ (if p ≤ q), or
δ_{|[0,t_{0}]} · d · δ_{|[t_{0},q]} and γ (if q ≤ p).

where c(t) = γ(t_{0}) and d(t) = δ(t_{0}) are the constant (τ-)paths with domain [0, |p − q|].

If 𝕃(G) denotes the set of all τ-loops that start and end at the identity element e_{G} of G, the restriction of ∼_{τ} to 𝕃(G) × 𝕃(G) is an equivalence relation on 𝕃(G). We define

π_{1}(G) := 𝕃(G)/ ∼_{τ}

and [γ] := the class of γ ∈ 𝕃(G). Note that π_{1}(G) is indeed a group with group operation given by [γ][δ] = [γ · δ].

In a similar way we define the o-minimal fundamental group π_{1}(X) of a definable set X ⊆ R^{n}.

Given the above two possible definitions of o-minimal fundamental groups it is natural to try to find out if they coincide. Our main result shows that this is the case:

Theorem 1.4 Let R be an o-minimal expansion of a group and G a definably t-connected definable group. Then the o-minimal universal covering homomorphism p̃ : G̃ ⟶ G is a locally definable covering homomorphism and π_{1}(G) is isomorphic to π(G).

Theorem 1.4 will actually be proved for definably τ-connected locally definable groups. See Theorem 3.11 below. As a consequence of our work we obtain the following corollary which is proved at the end of the paper.

Corollary 1.5 Let R be an o-minimal expansion of a group and G a definably t-connected definable group. Then π_{1}(G) is a finitely generated abelian group. Moreover, if G is abelian, then there is l ∈ ℕ such that π_{1}(G) ≃ ℤ^{l} and, for each k ∈ ℕ, the subgroup G[k] of k-torsion points of G is given by G[k] ≃ (ℤ/kℤ)^{l}.

When G is a definably compact, abelian definable group, we conjecture that l above is the dimension of G. This is known to be the case when R is linear ([5]) or R is an o-minimal expansion of a real closed field ([4]). So the conjecture is open for R eventually linear but not linear.

2 Preliminary results

This section contains all the lemmas that come from other references and are used later in the paper. Thus we generalize the theory of [3] and [4] Section 2 to the category of locally definable covering maps of locally definable groups in R. Since the arguments are similar we will omit the details.

Definition 2.1 A set Z is a locally definable set over A, where A ⊆ R and |A| < ℵ_{1}, if there is a countable collection {Z_{i} : i ∈ I} of definable subsets of R^{n}, all definable over A, such that: (i) Z = ∪{Z_{i} : i ∈ I}; (ii) for every i, j ∈ I there is k ∈ I such that Z_{i} ∪ Z_{j} ⊆ Z_{k}.

Given two locally definable sets X and Z over A, we say that X is a locally definable subset of Z over A if X is a subset of Z.

A map α : Z ⟶ X between locally definable sets over A is called a locally definable map over A if for every definable subset V ⊆ Z defined over A, the restriction α_{|V} is a definable map over A.

By saturation, the set Z does not depend on the choice of the collection {Z_{i} : i ∈ I}. Furthermore, if α : Z ⟶ X is a locally definable map over A between locally definable sets over A and Y is a locally definable subset of X over A, then the following hold:

(1) α(Z) is a locally definable subset of X over A and α^{−1}(Y) is a locally definable subset of Z over A.

(2) If Y is such that V ∩ Y is definable for every definable subset V of X, then W ∩ α^{−1}(Y) is definable for every definable subset W of Z. (Since W ∩ α^{−1}(Y) = α_{|W}^{−1}(α(W) ∩ Y))).

Definition 2.2 Let G be a locally definable group over A and W a locally definable set over A. A locally definable map w : W ⟶ G over A is called a locally definable covering map if w is surjective and there is a family {U_{l} : l ∈ L} of τ-open definable subsets of G over A such that G = ∪{U_{l} : l ∈ L} and, for each l ∈ L, the locally definable subset w^{−1}(U_{l}) of W over A is a disjoint union of definable subsets of W over A, each of which is mapped bijectively by w onto U_{l}.

We call {U_{l} : l ∈ L} a w-admissible family of definable τ-neighborhoods over A.

Given a locally definable covering map w : W ⟶ G over A there is a topology on W, which we call the w-topology, generated by the definable sets of the form w^{−1}(U) ∩ V, where U is a τ-open definable subset of G and V is one of the definable subsets of the disjoint union w^{−1}(U_{l}) for some U_{l} in the w-admissible family of definable τ-neighborhoods.

Clearly, with respect to the w-topology on W (and the τ-topology on G), w : W ⟶ G is continuous. Furthermore, w : W ⟶ G is an open surjection. In fact, let V be a w-open definable subset of W over A and, for each l ∈ L, let {U_{s}^{l} : s ∈ S_{l}} be the collection of w-open disjoint definable subsets of W over A such that w^{−1}(U_{l}) = ∪{U_{s}^{l} : s ∈ S_{l}} and w_{|U_{s}^{l}} : U_{s}^{l} ⟶ U_{l} is a definable homeomorphism over A for every s ∈ S_{l}. Since |A| < ℵ_{1}, by saturation, there is {W_{1}, …, W_{m}} ⊆ {U_{s}^{l} : l ∈ L, s ∈ S_{l}} such that V ⊆ ∪{W_{i} : i = 1, …, m}. But then V = ∪{V ∩ W_{i} : i = 1, …, m} and w(V) = ∪{w(V ∩ W_{i}) : i = 1, …, m} is τ-open.

Lemma 2.3 Let w : W ⟶ G be a locally definable covering map and suppose that W is also a locally definable group. Then on W the w-topology coincides with the τ-topology.

Proof. Let a ∈ W be a generic point and U a definable w-open neighborhood of a in W. We may assume that w_{|U} : U ⟶ w(U) is a definable homeomorphism. Since w(a) is also generic, there exists a definable subset V ⊆ w(U) containing w(a) such that V is both τ-open in G and open in G with the induced topology on G from R^{n}. Thus w^{−1}(V) is also both a wneighborhood of a in W and in W with the induced topology on W from R^{n}. Hence, w^{−1}(V) is a τ-neighborhood of a in W. By uniqueness of τ-topology, this implies that the w-topology and the τ-topology on W agree. □

Let w : W ⟶ G be a locally definable covering map (over some A with |A| < ℵ_{1}). Let X be a definable subset of W equipped with the induced w-topology from W. We will now introduce certain notions in parallel for X and W.

A w-path α : [0, p] ⟶ X (α : [0, p] ⟶ W) is a w-continuous definable map. A w-path α : [0, p] ⟶ X (α : [0, p] ⟶ W) is a w-loop if α(0) = α(p). A concatenation of two w-paths γ : [0, p] ⟶ X (γ : [0, p] ⟶ W) and δ : [0, q] ⟶ X (δ : [0, q] ⟶ W) with γ(p) = δ(0) is a w-path γ · δ : [0, p + q] ⟶ X (γ · δ : [0, p + q] ⟶ W) with:

(γ · δ)(t) = \{ \begin{matrix}γ(t) & if t ∈ [0, p] \\ δ(t − p) & if t ∈ [p, p + q]. \end{matrix}

Given two definable w-continuous maps f, g : Y ⊆ R^{m} ⟶ X (f, g : Y ⊆ R^{m} ⟶ W), we say that a definable w-continuous map F (t, s) : Y ×[0, q] ⟶ X (F (t, s) : Y × [0, q] ⟶ W) is a w-homotopy between f and g if f = F_{0} and g = F_{q}, where ∀s ∈ [0, q], F_{s} := F (·, s). In this situation we say that f and g are w-homotopic, denoted f ∼_{w} g.

Definition 2.4 Two w-paths γ : [0, p] ⟶ X (γ : [0, p] ⟶ W), δ : [0, q] ⟶ X (δ : [0, q] ⟶ W), with γ(0) = δ(0) and γ(p) = δ(q), are called whomotopic if there is some t_{0} ∈ [0, min{p, q}], and a w-homotopy F (t, s) : [0, max{p, q}] × [0, r] ⟶ X (F (t, s) : [0, max{p, q}] × [0, r] ⟶ W), for some r > 0 in R, between

γ_{|[0,t_{0}]} · c · γ_{|[t_{0},p]} and δ (if p ≤ q), or
δ_{|[0,t_{0}]} · d · δ_{|[t_{0},q]} and γ (if q ≤ p).

where c(t) = γ(t_{0}) and d(t) = δ(t_{0}) are the constant w-paths with domain [0, |p − q|].

If 𝕃(W) denotes the set of all w-loops that start and end at a fixed element e_{W} of W such that w(e_{W}) = e_{G}, the restriction of ∼_{w} to 𝕃(W) × 𝕃(W) is an equivalence relation on 𝕃(W). We define

π_{1}(W) := 𝕃(W)/ ∼_{w}

and [γ] := the class of γ ∈ 𝕃(W). Note that π_{1}(W) is indeed a group with group operation given by [γ][δ] = [γ · δ]. Also this group depends on the w-topology on W.

In a similar way we define the o-minimal fundamental group π_{1}(X) of a definable subset X ⊆ W with respect to the induced w-topology.

Clearly, any two constant w-loops at the same point c ∈ W are whomotopic. We will thus write ϵ_{c} for the constant w-loop at c without specifying its domain.

In view of Lemma 2.3, we obtain the above notions with w replaced by τ for definable subsets of a locally definable group equipped with the induced τ-topology.

Lemma 2.5 Let w : W ⟶ G and v : V ⟶ H be locally definable covering maps. Then (w, v) : W × V ⟶ G × H is a locally definable covering map and θ : π_{1}(W) × π_{1}(V) ⟶ π_{1}(W × V) : ([γ], [δ]) ↦ [(γ, δ)] is a group isomorphism.

Proof. The inverse of θ is π_{1}(W × V) ⟶ π_{1}(W) × π_{1}(V) : [ρ] ↦ ([q_{1} ∘ ρ], [q_{2} ∘ ρ]) where q_{1} and q_{2} are the projections from W × V onto W and V, respectively. □

Let w : W ⟶ G be a locally definable covering map (over some A with |A| < ℵ_{1}). Let Z be a definable set and let f : Z ⟶ G be a definable continuous map (with respect to the τ-topology on G). A lifting of f is a continuous definable map f̃: Z ⟶ W (with respect to the w-topology on W) such that p ∘ f̃ = f.

Lemma 2.6 Let w : W ⟶ G be a locally definable covering map, Z a definably connected definable set and f : Z ⟶ G a definable continuous map. If f̃_{1}, f̃_{2} : Z ⟶ W are two liftings of f, then f̃_{1} = f̃_{2} provided there is a z ∈ Z such that f̃_{1}(z) = f̃_{2}(z).

Proof. As in the proof of [3] Lemma 3.2, both sets {w ∈ Z : f̃_{1}(w) = f̃_{2}(w)} and {w ∈ Z : f̃_{1}(w) ≠ f̃_{2}(w)} are definable and open, the first one is nonempty. □

Lemma 2.7 Suppose that w : W ⟶ G is a locally definable covering map. Then the following hold.

(1) Let γ be a τ-path in G and y ∈ W. If w(y) = γ(0), then there is a unique w-path γ̃ in W, lifting γ, such that γ̃(0) = y.

(2) Suppose that F is a τ-homotopy between the τ-paths γ and σ in G. Let γ̃ be a w-path in W lifting γ. Then there is a unique definable lifting F̃ of F, which is a w-homotopy between γ̃ and σ̃, where σ̃ is a w-path in W lifting σ.

Proof. In our category, the path and the homotopy liftings can be proved as in [4] by observing that, by saturation, a definable subset of G is covered by finitely many open definable subsets of G. □

Notation: Referring to Lemma 2.7, if γ : [0, q] ⟶ G is a τ-path in G and y ∈ W, we denote by y ∗ γ the final point γ̃(q) of the lifting γ̃ of γ with initial point γ̃(0) = y.

The following consequence of Lemma 2.7 is proved in exactly the same way as its definable analogue in [4] Corollary 2.9. Below, for w : W ⟶ G a locally definable covering map, we say that W is definably w-connected if there is no proper locally definable subset of W which is both w-open and wclosed and whose intersection with any definable subset of W is definable. In view of Lemma 2.3, this notion generalizes the notion of definably connected in locally definable groups studied in [3].

Remark 2.8 Suppose that w : W ⟶ G is a locally definable covering map and let y ∈ W be such that w(y) = e_{G}. Suppose that W and G are definably w-connected and τ-connected respectively. Then we have a well defined homomorphism w_{∗} : π_{1}(W) ⟶ π_{1}(G) : [γ] ↦ [w ∘ γ] and the following hold.

(1) If σ is a τ-path in G from e_{G} to e_{G}, then y = y ∗ σ if and only if [σ] ∈ w_{∗}(π_{1}(W)).

(2) If σ and σ′ are two τ-paths in G from e_{G} to x, then y ∗ σ = y ∗ σ′ if and only if [σ · σ^{′−1}] ∈ w_{∗}(π_{1}(W)).

Let w : W ⟶ G be a locally definable covering map. We say that W is w-path connected if for every u, v ∈ W there is a w-path α : [0, q] ⟶ W such that α(0) = u and α(q) = v.

Lemma 2.9 Let w : W ⟶ G be a locally definable covering map. Then W is definably w-connected if and only if W is w-path connected. In fact, for any definably w-connected definable subset X of W there is a uniformly definable family of w-paths in X connecting a given fixed point in X to any other point in X.

Proof. Since w : W ⟶ G is a locally definable covering map, it is enough to prove the result for locally definable groups. By the first part of the proof of [9] Lemma 2.13, there is a locally definable subset U of G such that dim(G \ U) < dimG, the intersection of any definable subset of G with U is a definable subset and the induced τ-topology on U coincides with the induced topology from R^{n}. So a definable subset B of U is τ-connected if and only if B is definably connected (in R^{n}). Thus the result follows from by [6] Chapter VI, Proposition 3.2 and its proof, saturation and [3] Lemma 3.5 (i.e., countably many translates of U cover G). □

The next proposition is also a consequence of Lemma 2.7 and is proved in exactly the same way as its definable analogue in [4] Corollary 2.8 and Proposition 2.10.

Proposition 2.10 Let w : W ⟶ G be a locally definable covering map. Suppose that W and G are definably w-connected and τ-connected respectively. Then the following hold:

(1) w_{∗} : π_{1}(W) ⟶ π_{1}(G) is an injective homomorphism;

(2) π_{1}(G)/w_{∗}(π_{1}(W)) ≃ Aut(W/G) (the group of all locally definable whomeomorphisms ϕ : W ⟶ W such that w = w ∘ ϕ).

Below we will also require the following generalization of Lemma 2.6:

Lemma 2.11 Let w : W ⟶ G and v : V ⟶ H be locally definable covering maps and let f, g : V ⟶ W be two continuous locally definable maps (with respect to the v and w topologies) such that w ∘ f = w ∘ g. If V is definably v-connected and f(x) = g(x) for some x ∈ V, then f = g.

Proof. This is as in [3] Lemma 3.2 once we show that {x ∈ V : f(x) = g(x)}, which is open and closed, is a locally definable subset whose intersection with any definable subset of V is a definable subset of V. If C, D ⊆ V are definable, then (V ×_{W} V) ∩ (C × D) = {(x, y) ∈ C × D : f_{|C}(x) = g_{|D}(y)} is definable, and so (V ×_{W} V) ∩ E is definable for every definable subset E of V × V. Similarly, Δ_{V} ∩ E is definable for every definable subset E of V × V. Hence, (V ×_{W} V) ∩ Δ_{V} ∩ E is definable for every definable subset E of V × V. From this and the observation (2) on page 5 we get our result since {x ∈ V : f(x) = g(x)} = i^{−1}((V ×_{W} V) ∩ Δ_{V}), where i : V ⟶ Δ_{V} : x ↦ (x, x) is a locally definable map. □

Finally we include the following result ([3] Proposition 3.4) which will also be useful later:

Proposition 2.12 Let h : H ⟶ G be a locally definable covering homomorphism and suppose that H is definably τ-connected. Then

Kerh ≃ Aut(H/G)

and Aut(H/G) is abelian.

3 The universal covering homomorphism

Here we will present the proof of our main result. We start however with a special case.

3.1 A special case of the main result

The main result of the paper [5], in the language of the theory of locally definable covering homomorphisms, is the following (compare with [5] Remark 6.14). For a related result see also [8].

Theorem 3.1 ([5]) Suppose that R is an ordered vector space over an ordered division ring and G is a definably t-connected, definably compact, definable group of dimension n. Then there is a locally definable group V which is a subgroup of (R^{n}, +) and a locally definable covering homomorphism v : V ⟶ G such that π_{1}(G) ≃ Kerv ≃ ℤ^{n}.

In [5] Remark 6.14 it is suggested that v : V ⟶ G is in some sense the universal cover of G since we have π_{1}(V) = 1 ([5] Corollary 6.7). This claim can now be made more precise:

Theorem 3.2 Suppose that R is an ordered vector space over an ordered division ring and G is a definably t-connected, definably compact, definable group of dimension n. Then the locally definable covering homomorphism v : V ⟶ G is isomorphic to p̃: G̃ ⟶ G and π_{1}(G) ≃ π(G) ≃ ℤ^{n}.

Proof. Suppose that q : K ⟶ V is a locally definable covering homomorphism. Then from Propositions 2.10 and 2.12 we obtain Kerq ≃ Aut(K/V) ≃ π_{1}(V)/q_{∗}(π_{1}(K)) = 1 since π_{1}(V) = 1, by [5] Corollary 6.7. So q : K ⟶ V is a locally definable isomorphism (since it is surjective). Consequently, by [3] Lemma 3.8, the set of all h : H ⟶ G in Cov^{0}(G) which are locally definably isomorphic to v : V ⟶ G is cofinal in Cov^{0}(G) and hence the inverse limit p̃: G̃ ⟶ G is isomorphic to v : V ⟶ G. By Propositions 2.10 and 2.12 we obtain π(G) ≃ Kerv ≃ Aut(V/G) ≃ π_{1}(G) since π_{1}(V) = 1. Thus the result holds as required. □

3.2 The main result

Here we prove the main result of the paper. Before we proceed we need the following propositions.

Proposition 3.3 Let G be a definably τ-connected locally definable group of dimension k. Then there is a countable collection {O_{s} : s ∈ S} of τ-open definably τ-connected definable subsets of G with G = ∪{O_{s} : s ∈ S} and, for each s ∈ S, O_{s} is definably homeomorphic to an open cell in R^{k}. In particular, for each s ∈ S, the o-minimal fundamental group π_{1}(O_{s}) with respect to the induced τ-topology on O_{s} is trivial

Proof. By the first part of the proof of [9] Lemma 2.13, there is a locally definable subset U of G such that dim(G\U) < dimG, the intersection of any definable subset of G with U is a definable subset and the induced τ-topology on U coincides with the induced topology from R^{n}. Without loss of generality we can assume that U is a countable union of cells of dimension k = dimG. Note that on each of these k-cells in U, the induced τ-topology coincides with the induced topology from R^{n}. By [3] Lemma 3.5 countably many translates of U cover G, so countably many τ-open definably τ-connected subsets of G which are definably τ-homeomorphic to k-cells in U cover G.

Let {O_{s} : s ∈ S} be this collection. To finish, it is enough to show that if C is an open cell in R^{k} then π_{1}(C) = 1 (since definable homeomorphisms induce isomorphisms between the o-minimal fundamental groups).

We will show this by induction on the construction of cells. If C has dimension zero then this is obvious. Assume that C = (a, b) ⊆ R∪{−∞, +∞} is an open cell of dimension one and α : [0, q] ⟶ C is a definable loop at c ∈ C. Consider the continuous definable map H : [0, q] × [0, q] ⟶ C given by

H(t, x) := α(\frac{t + x + |t − x|}{2}).

Then H is a definable homotopy between α and ϵ_{c}. So [α] = 1 and π_{1}(C) = 1 as required.

Suppose that B is a cell, π_{1}(B) = 1 and C = (f, g)_{B} with f, g : B ⟶ R∪ {−∞, +∞} continuous definable maps such that f < g. Let c = (b, a) ∈ C and let σ : [0, q] ⟶ C be a definable loop at c. We can write σ(t) = (β(t), α(t)) for some definable loop β : [0, q] ⟶ B at b and α : [0, q] ⟶ R a definable loop at a. By assumption there is a definable homotopy F : [0, q] × [0, p] ⟶ B between β and ϵ_{b} and a definable homotopy E : [0, q] × [0, r] ⟶ R between α and ϵ_{a}. Let H : [0, q] × [0, max{r, p}] ⟶ C be the definable map such that if r ≤ p then

H(t, x) = \{ \begin{matrix}(F (t, x), E(t, x)) & if x ≤ r, \\ (F (t, x), E(t, r)) & if x ≥ r, \end{matrix}

and if p ≤ r then

H(t, x) = \{ \begin{matrix}(F (t, x), E(t, x)) & if x ≤ p, \\ (F (t, p), E(t, x)) & if x ≥ p. \end{matrix}

Then H is a definable homotopy between σ and ϵ_{c}. So [σ] = 1 and π_{1}(C) = 1 as required. □

Proposition 3.4 Let G be a definably τ-connected locally definable group. Then the o-minimal fundamental group π_{1}(G) of G (with respect to the induced τ-topology) is countable. In fact, if G is definable, then π_{1}(G) is finitely generated.

Proof. Consider the countable cover {O_{s} : s ∈ S} of G by τ-open definably τ-connected definable subsets given by Proposition 3.3. For each pair of distinct elements s, t ∈ S such that O_{s} ∩O_{t} ≠ ∅ and for each definably τ-connected component C of this intersection choose a point a_{s,t,C} ∈ C. For each pair (a_{s,t,C}, a_{s′,t′,D}) of distinct points and l ∈ {s, t} ∩ {s′, t′} let σ_{(C,D),s,t,s′,t′}^{l} be a τ-path in O_{l} from a_{s,t,C} to a_{s′,t′,D}. Also, for each a_{s,t,C} such that e_{G} ∈ O_{s}, let σ_{(e_{G},C),s,t}^{s} (respectively, σ_{(C,e_{G}),s,t}^{s}) be a τ-path in O_{s} from e_{G} to a_{s,t,C} (respectively, from a_{s,t,C} to e_{G}).

Let Σ be the countable collection of all τ-paths σ_{(C,D),s,t,s′,t′}^{l}, σ_{(e_{G},C),s,t}^{s} and σ_{(C,e_{G}),s,t}^{s} as above. The set Σ generates a free countable language Σ^{∗} such that some of its words correspond in an obvious way to τ-paths in G. To finish the proof it is enough to show that any τ-loop in G is τ-homotopic to a τ-loop which is a concatenation of τ-paths in Σ and thus corresponds to a word in Σ^{∗}.

Let λ be a τ-loop in G. Then by saturation and o-minimality there exists a minimal k for which we can choose points 0 = t(0) < t(1) < ⋯ < t(k) < t(k+1) = q_{λ} such that for each j = 0, …, k, we have λ([t(j), t(j+1)]) ⊆ O_{s(j)} for some s(j) ∈ S. Thus λ = λ_{0} · · · · · λ_{k} where, for each j, λ_{j} : [0, q_{λ_{j}}] ⟶ G is the τ-path with q_{λ_{j}} = t(j + 1) − t(j) and given by λ_{j}(t) = λ(t + t(j)). For i = 0, …, k − 1, let C_{i} be the definably τ-connected component of O_{s(i)} ∩ O_{s(i+1)} containing λ_{i}(q_{λ_{i}}) and let ϵ_{i} be a τ-path in C_{i} from a_{s(i),s(i+1),C_{i}} to λ_{i}(q_{λ_{i}}). Let σ_{0} be the τ-path σ_{(e_{G},C_{0}),s(0),s(1)}^{s(0)} in O_{s(0)} and let σ_{k} be the τ-path

σ_{(C_{k−1},e_{G}),s(k−1),s(k)}^{s(k)} in O_{s(k)}. Finally, for i = 1, …, k − 1, let σ_{i} be the τ-path

σ_{(C_{i−1},C_{i}),s(i−1),s(i),s(i),s(i+1)}^{s(i)} in O_{s(i)}. Since by Proposition 3.3, π_{1}(O_{s(j)}) = 1 for all j = 0, …, k, we have that σ_{0} is τ-homotopic to λ_{0} · ϵ_{0}^{−1}, σ_{k} is τ-homotopic to ϵ_{k−1} · λ_{k} and, for each i = 1, …, k − 1, σ_{i} is τ-homotopic to ϵ_{i−1} · λ_{i} · ϵ_{i}^{−1}. Hence, λ is τ-homotopic to σ_{0} · σ_{1} · · · · · σ_{k} ∈ Σ^{∗} as required.

Assume now that G is definable. Let K be the simplicial complex of dimension one whose vertices are the end points of the τ-paths in Σ and whose edges are the images of the τ-paths in Σ. Clearly we have a homomorphism π_{1}(|K|, e_{G}) ⟶ π_{1}(G) which sends an edge loop in K into the τ-loop it determines in G. This is well defined since if two edge loops are homotopic in |K| then they are obviously τ-homotopic in G. The argument in the previous paragraph shows that the homomorphism π_{1}(|K|, e_{G}) ⟶ π_{1}(G) is surjective. Now as explained in [2] Chapter 3, Subsection 3.5.3, the fundamental group of a (finite) simplicial complex is finitely generated. Hence π_{1}(G) is also finitely generated. □

For the rest of the section, fix G a definably τ-connected locally definable group.

We will construct now an “abstract universal covering map” u : U ⟶ G from which we will obtain a locally definable covering map v : V ⟶ G which will be a locally definable covering homomorphism once we put a suitable locally definable group structure on V. The later will then be shown to be isomorphic to p̃: G̃ ⟶ G.

Given two τ-paths σ : [0, q_{σ}] ⟶ G and λ : [0, q_{λ}] ⟶ G in G, we put σ ≃ λ if and only if σ(0) = λ(0) = e_{G}, σ(q_{σ}) = λ(q_{λ}) and [σ · λ^{−1}] = 1 ∈ π_{1}(G). Here, λ^{−1} : [0, q_{λ^{−1}}] ⟶ G is the τ-path such that q_{λ^{−1}} = q_{λ} and λ^{−1}(t) = λ(q_{λ} − t) for every t in [0, q_{λ^{−1}}]. The relation ≃ is an equivalence relation and we denote the equivalence class of σ under ≃ by ⟨σ⟩. For each s ∈ S, let U_{s} = {⟨σ⟩ : σ is a τ-path in G such that σ(0) = e_{G} and σ(q_{σ}) ∈ O_{s}} and fix a τ-path σ_{s} : [0, q_{s}] ⟶ G such that σ(0) = e_{G} and σ(q_{s}) ∈ O_{s}.

Claim 3.5 There is a well-defined bijection

ϕ_{s} : U_{s} ⟶ O_{s} × π_{1}(G) : ⟨λ⟩ ↦ (λ(q_{λ}), [λ · η · σ_{s}^{−1}]),

where η : [0, q_{η}] ⟶ O_{s} is a τ-path in O_{s} such that η(0) = λ(q_{λ}) and η(q_{η}) = σ_{s}(q_{s}).

Proof. Clearly, ϕ_{s} is well-defined, i.e. it does not depend on the choice of η since π_{1}(O_{s}) = 1 (Proposition 3.3) and for ⟨λ⟩ = ⟨λ′⟩ we have λ(q_{λ}) = λ(q_{λ′}) and

[λ · η · σ_{s}^{−1}] = [λ · λ^{′−1} · λ′ · η · σ_{s}^{−1}]
= [λ · λ^{′−1}][λ′ · η · σ_{s}^{−1}]
= [λ′ · η · σ_{s}^{−1}].

Also, for o ∈ O_{s} and [γ] ∈ π_{1}(G) we have ϕ_{s}(⟨λ⟩) = (o, [γ]) for λ = γ · σ_{s} · η^{−1}, where η : [0, q_{η}] ⟶ G is a τ-path in O_{s} such that η(0) = o and η(q_{η}) = σ_{s}(q_{s}). Thus ϕ_{s} is surjective. On the other hand, suppose that ϕ_{s}(⟨λ⟩) = ϕ_{s}(⟨λ′⟩). Then λ(q_{λ}) = λ′(q_{λ′}) and [λ · η · σ_{s}^{−1}] = [λ′ · η′ · σ_{s}^{−1}]. But we also have

[λ · η · σ_{s}^{−1}] = [λ · λ^{′−1}][λ′ · η · σ_{s}^{−1}]
= [λ · λ^{′−1}][λ′ · η′ · σ_{s}^{−1}]
= [λ · λ^{′−1}][λ · η · σ_{s}^{−1}]

(the fact π_{1}(O_{s}) = 1 (Proposition 3.3) implies that λ′ · η · σ_{s}^{−1} is τ-homotopic to λ′ · η′ · σ_{s}^{−1}). Thus we have [λ · λ^{′−1}] = 1, ⟨λ⟩ = ⟨λ′⟩ and ϕ_{s} is injective. □

Set U = ∪{U_{s} : s ∈ S} and let u : U ⟶ G be the surjective map given by u(⟨λ⟩) = λ(q_{λ}). By Claim 3.5 and its proof we have, for each s ∈ S,

(•) u^{−1}(O_{s}) is the disjoint union of the subsets ϕ_{s}^{−1}(O_{s} × {[γ]}) with [γ] ∈ π_{1}(G); (••) u restricted to ϕ_{s}^{−1}(O_{s} × {[γ]}) is a bijection onto O_{s}.

Claim 3.6 If s, t ∈ S are such that O_{s} ∩ O_{t} ≠ ∅ and C is a definably τconnected component of O_{s} ∩ O_{t}, then the restriction of the bijection

ϕ_{t} ∘ ϕ_{s}^{−1} : (O_{s} ∩ O_{t}) × π_{1}(G) ⟶ (O_{s} ∩ O_{t}) × π_{1}(G)

to C × {[γ]} is the same as C × {[γ]} ⟶ C × {[γ_{C}]} : (o, [γ]) ↦ (o, [γ_{C}]) for some [γ_{C}] ∈ π_{1}(G).

Proof. Let o ∈ C. By Claim 3.5 and its proof , ϕ_{t}∘ϕ_{s}^{−1}(o, [γ]) = (o, [λ·η′· σ_{t}^{−1}]), where λ = γ ·σ_{s} ·η^{−1} and η : [0, q_{η}] ⟶ O_{s} and η′ : [0, q_{η′}] ⟶ O_{t} are τpaths such that η(0) = η′(0) = o, η(q_{η}) = σ_{s}(q_{s}) and η′(q_{η′}) = σ_{t}(q_{t}). Thus to prove the claim it is enough to show that [γ·σ_{s}·η^{−1}·η′·σ_{t}^{−1}] = [γ·σ_{s}·θ^{−1}·θ′·σ_{t}^{−1}] whenever θ : [0, q_{θ}] ⟶ O_{s} and θ′ : [0, q_{θ′}] ⟶ O_{t} are τ-paths such that θ(0) = θ′(0) ∈ C, θ(q_{θ}) = σ_{s}(q_{s}) and θ′(q_{θ′}) = σ_{t}(q_{t}).

Since C is τ-path connected, let ρ : [0, q_{ρ}] ⟶ C be a τ-path such that ρ(0) = o and ρ(q_{ρ}) = θ(0) = θ′(0). Now using the fact that π_{1}(O_{s}) = π_{1}(O_{t}) = 1 (Proposition 3.3) we see that ρ · θ (respectively θ′ · ρ^{−1}) is τhomotopic to η (respectively η^{′−1}). Thus η^{−1} · η′ is τ-homotopic to θ^{−1} · θ′. From here we get [γ · σ_{s} · η^{−1} · η′ · σ_{t}^{−1}] = [γ · σ_{s} · θ^{−1} · θ′ · σ_{t}^{−1}] as required. □

We will let 1 ∈ R be a fixed 0-definable positive element of R and denote the element n · 1 of the group (R, 0, +) by n. By Proposition 3.4, we will identify π_{1}(G) with a subset of ℕ ⊆ R and thus, assuming that G ⊆ R^{l},

O_{(s,[γ])} := O_{s} × {[γ]}

is a definable subset of R^{l+1} and O := ∪{O_{(s,[γ])} : (s, [γ]) ∈ S × π_{1}(G)} is a locally definable subset of R^{l+1}.

Let {(s_{i}, l_{j}) : (i, j) ∈ ℕ × ℕ} be an enumeration of S × π_{1}(G). Define inductively (on i) the sets N_{i}, O′_{(s_{i},l_{j})} and V_{(s_{i},l_{j})} in the following way:

N_{0} = ∅ and O′_{(s_{0},l_{j})} = V_{(s_{0},l_{j})} = O_{(s_{0},l_{j})};

assuming that N_{i}, O′_{(s_{i},l_{j})} and V_{(s_{i},l_{j})} were already defined, put

N_{i+1} = {n : n < i + 1 and O_{s_{i+1}} ∩ O_{s_{n}} ≠ ∅};

O′_{(s_{i+1},l_{j})} = O_{(s_{i+1},l_{j})} \ ∪{C × {l_{j}} : C is a definably τ-connected component of O_{s_{i+1}} ∩ O_{s_{n}}, n ∈ N_{i+1} and (ϕ_{s_{i+1}} ∘ ϕ_{s_{n}}^{−1})_{|C×{l_{C}}}(o, l_{C}) = (o, l_{j})};

V_{(s_{i+1},l_{j})} = O′_{(s_{i+1},l_{j})} ∪ ⋃{V_{(s_{n},l_{C})}^{C} : C is a definably τ-connected component of O_{s_{i+1}} ∩ O_{s_{n}}, n ∈ N_{i+1} and (ϕ_{s_{i+1}} ∘ ϕ_{s_{n}}^{−1})_{|C×{l_{C}}}(o, l_{C}) = (o, l_{j})}, where V_{(s_{n},l_{C})}^{C} = {x ∈ V_{(s_{n},l_{C})} : x = (o, l) with o ∈ C}. ^{1}

1We wish to thank here Elias Baro (Universidad Autónoma de Madrid) for pointing out an imprecision on an early version of our inductive construction.

By Claim 3.6, the sets V_{(s_{i},l_{j})} are well defined definable subsets of R^{l+1}.

Claim 3.7 Let V = ∪{V_{(s_{i},l_{j})} : (i, j) ∈ ℕ × ℕ}. Then V is a locally definable set and the surjective map v : V ⟶ G given by the projection onto the first coordinate is a locally definable covering map, i.e., for each i, we have:

(1) v^{−1}(O_{s_{i}}) = ∪{V_{(s_{i},l_{j})} : j ∈ ℕ} (disjoint union);

(2) v_{|V_{(s_{i},l_{j})}} is a definable bijection onto O_{s_{i}}.

Proof. This follows by induction on the definition of the definable sets V_{(s_{i},l_{j})} together with Claim 3.6. □

Fix s_{e_{G}} ∈ S such that e_{G} ∈ O_{s_{e_{G}}} and assume without loss of generality that σ_{s_{e_{G}}} = ϵ_{e_{G}} (the trivial τ-loop at e_{G}, see page 7). Let e_{V} = (e_{G}, [ϵ_{e_{G}}]) ∈ V.

Claim 3.8 Let (o, [γ]) ∈ V and suppose that λ : [0, q_{λ}] ⟶ G is a τ-path such that λ(0) = e_{G}, λ(q_{λ}) = o and ϕ_{s}(⟨λ⟩) = (o, [γ]). Then there exists a vpath λ̃ : [0, qλ̃] ⟶ V in V such that λ̃(0) = e_{V}, λ̃(qλ̃) = (o, [γ]) and v∘λ̃ = λ. In particular, V is v-path connected and the o-minimal fundamental group π_{1}(V) of V with respect to the v-topology is trivial.

Proof. By saturation and o-minimality there exists a minimal k for which we can choose points 0 = t(0) < t(1) < ⋯ < t(k) < t(k + 1) = q_{λ} such that for each j = 0, …, k, we have λ([t(j), t(j + 1)]) ⊆ O_{s(j)} for some s(j) ∈ S.

We prove the result by induction on k. If k = 0, then λ([0, q_{λ}]) ⊆ O_{s(0)} and [γ] = [ϵ_{e_{G}}], and we put λ̃ := (v_{|V_{(s(0),[ϵe_{G}])}})^{−1} ∘ λ. For the inductive step let η := λ_{|[0,t(k)]} and δ : [0, q_{λ} − t(k)] ⟶ O_{s(k)} : t ↦ λ(t + t(k)). By the induction hypothesis, let η̃: [0, t(k)] ⟶ V be a v-path such that η̃(0) = e_{V}, η̃(t(k)) = (η(t(k)), [γ′]) and v ∘ η̃ = η, where ϕ_{s(k−1)}(⟨η⟩) = (η(t(k)), [γ′]). Assume that s(k) appear after s(k − 1) in the enumeration of S introduced before. The other case is treated symmetrically. If ϕ_{s(k)}(⟨η⟩) = (η(t(k)), [γ′′]), then (η(t(k)), [γ′]) and (η(t(k)), [γ′′]) are the same point in V_{(s(k),[γ′′])}. Since λ = η · δ and π_{1}(O_{s(k)}) = 1 (Proposition 3.3), we have [γ] = [γ′′]. Thus, if δ̃:= (v_{|V_{(s(k),[γ′′])}})^{−1} ∘ δ, then η̃(t(k)) = δ̃(0), and λ̃ := η̃ · δ̃ satisfies the claim. So, in particular, V is v-path connected.

By Lemma 2.7, any v-loop δ in V at e_{V} is the unique lifting λ̃ of a τloop λ = v ∘ δ in G at e_{G} as defined in the previous paragraph. So we see that (e_{G}, [ϵ_{e_{G}}]) = e_{V} = λ̃(0) and e_{V} = λ̃(qλ̃) = (e_{G}, [λ]). This implies that [λ] = 1 and so v_{∗}([λ̃]) = [λ] = 1. Therefore, since by Proposition 2.10 (i), v_{∗} : π_{1}(V) ⟶ π_{1}(G) is injective, it follows that π_{1}(V) = 1. □

Our next goal is to make the locally definable covering map v : V ⟶ G into a locally definable covering homomorphism. For this we will need the following claim:

Claim 3.9 Let h : Y ⟶ X be either v : V ⟶ G or (v, v) : V × V ⟶ G × G, and let e_{Y} be e_{V} or (e_{V} , e_{V}) respectively, and e_{X} be e_{G} or (e_{G}, e_{G}) respectively. Suppose that g : X ⟶ G is a continuous locally definable map such that g(e_{X}) = e_{G}. Then there is a unique continuous locally definable map g̃: Y ⟶ V such that g̃(e_{Y}) = e_{V} and v ∘ g̃ = g ∘ h.

Proof. The uniqueness of such a locally definable lifting g̃ of g ∘ h follows from Lemma 2.11. To construct g̃ : Y ⟶ V we will use the fact that h : Y ⟶ X is a locally definable covering map, and by Lemma 2.5 and Claim 3.8, π_{1}(V × V) ≃ π_{1}(V) × π_{1}(V) = 1. We will also use the notation introduced right after Lemma 2.7.

Let {U_{l} : l ∈ L} be either {O_{s} : s ∈ S} or {O_{s} × O_{t} : s, t ∈ S}. Let f = g ∘ h : Y ⟶ G and for each l ∈ L, let {V_{i}^{l} : i ∈ I_{l}} be the definably h-connected components of f^{−1}(U_{l}). For all l ∈ L, i ∈ I_{l}, choose y_{i}^{l} ∈ V_{i}^{l} such that if e_{Y} ∈ V_{i}^{l} then e_{Y} = y_{i}^{l}, and let η_{i}^{l} be an h-path in Y from e_{Y} to y_{i}^{l}. Since each V_{i}^{l} is definably h-connected, by Lemma 2.9 there is a uniformly definable family {γ_{i}^{l}(w) : w ∈ V_{i}^{l}} of h-paths in V_{i}^{l} from y_{i}^{l} to w. For w ∈ V_{i}^{l}, let δ_{i}^{l}(w) be the h-path η_{i}^{l} · γ_{i}^{l}(w) from e_{Y} to w. Let σ_{i}^{l}(w) = f ∘ δ_{i}^{l}(w) and put f̃(w) = e_{Y} ∗ σ_{i}^{l}(w).

If w ∈ V_{i}^{l} ∩ V_{j}^{k} then we have another h-path δ_{j}^{k}(w) from e_{Y} to w obtained from V_{j}^{k}, and f ∘ (δ_{j}^{k}(w) · (δ_{i}^{l}(w))^{−1}) = σ_{j}^{k}(w) · (σ_{i}^{l}(w))^{−1} is a τ-path from e_{G} to e_{G}. By hypothesis, [σ_{j}^{k}(w) · (σ_{i}^{l}(w))^{−1}] ∈ f_{∗}(π_{1}(Y)) = 1 and by Remark 2.8 (2), e_{Y} ∗ σ_{i}^{l}(w) = e_{Y} ∗ σ_{j}^{k}(w) and so f̃ is well defined. Note that the same argument shows that f̃ does not depend on the choice of the points y_{i}^{l} ∈ V_{i}^{l} or of the h-paths η_{i}^{l}.

We now show that f̃ is a locally definable map. For this it is enough to show that f̃_{|V_{i}^{l}} is a definable map since by saturation any definable subset of Y is contained in a finite union of V_{i}^{l}’s. But for w ∈ V_{i}^{l}, we have f̃(w) = e_{Y} ∗ σ_{i}^{l}(w) which is the endpoint of the lifting σ_{j}^{l}(w)^{}of σ_{j}^{l}(w) starting at e_{Y}.

Since σ_{j}^{l}(w) = (f ∘η_{i}^{l})·(f ∘γ_{i}^{l}(w)), f̃(w) is the endpoint of the lifting f ∘ γ_{i}^{l}(w)^{}

of f ∘ γ_{i}^{l}(w) starting at the endpoint f ∘ η_{i}^{l}^{}(q_{η_{i}^{l}}) of the lifting f ∘ η_{i}^{l}^{}of f ∘ η_{i}^{l}. Thus, if W_{i}^{l} is a v-open subset of v^{−1}(O_{l}) such that v_{|W_{i}^{l}} : W_{i}^{l} ⟶ O_{l} is a

definable homeomorphism and f ∘ η_{i}^{l}^{}(q_{η_{i}^{l}}) ∈ W_{i}^{l}, then f̃(w) = ((v_{|W_{i}^{l}})^{−1} ∘ (f ∘ γ_{i}^{l}(w)))(q_{γ_{i}^{l}(w)}) where q_{γ_{i}^{l}(w)} is the end point of the domain of γ_{i}^{l}(w).

To finish we need to show that g̃ := f̃ is continuous. For this we use v ∘ g̃ = g ∘ h = f (which is immediate from the above characterization of f̃(w)) and the fact that, as remarked after Definition 2.2, v : V ⟶ G is an open mapping. □

Let μ : G × G ⟶ G and ι : G ⟶ G be the multiplication and the inverse in G. Let μ̃ : V × V ⟶ V and ι̃: V ⟶ V be the unique continuous locally definable maps given by Claim 3.9.

Claim 3.10 (V, μ̃, ι̃, e_{V}) is a locally definable group and v : V ⟶ G is a locally definable covering homomorphism.

Proof. We have that μ̃ ∘ (μ̃ × id_{V}) and μ̃ ∘ (id_{V} × μ̃) are the liftings of the same continuous locally definable map μ ∘ (μ × id_{G}) = μ ∘ (id_{G} × μ) and they coincide at (e_{V} , e_{V} , e_{V}). Thus by Lemma 2.11, we have μ̃ ∘ (μ̃ × id_{V}) = μ̃ ∘ (id_{V} × μ̃) and so (V, μ̃) is a locally definable semigroup. Similarly, we see that μ̃ ∘ (ι̃× id_{V}) ∘ Δ_{V} = e_{V} = μ̃ ∘ (id_{V} × ι̃) ∘ Δ_{V} and μ̃ ∘ i_{1}^{V} = id_{V} = μ̃ ∘ i_{2}^{V} where Δ_{V} : V ⟶ V ×V is the diagonal map, i_{1}^{V} : V ⟶ V ×V : v ↦ (v, e_{V}) and i_{2}^{V} : V ⟶ V × V : v ↦ (e_{V} , v). Thus (V, μ̃, ι̃, e_{V}) is a locally definable group as required. Since v ∘ μ̃ = μ ∘ (v, v) and v ∘ ι̃ = ι ∘ v, it follows that v : V ⟶ G is a locally definable homomorphism which must be a locally definable covering homomorphism since it is also a locally definable covering map. □

We are now ready to prove the main theorem of the paper (Theorem 1.4 in the introduction is a special case of this):

Theorem 3.11 Let G be a definably τ-connected locally definable group. Then the o-minimal universal covering homomorphism p̃ : G̃ ⟶ G is a locally definable covering homomorphism and π_{1}(G) is isomorphic to π(G).

Proof. Suppose that q : K ⟶ V is a locally definable covering homomorphism. Then from Propositions 2.10 and 2.12 we obtain Kerq ≃ Aut(K/V) ≃ π_{1}(V)/q_{∗}(π_{1}(K)) = 1 since π_{1}(V) = 1, by Claim 3.8. So q : K ⟶ V is a locally definable isomorphism (since it is surjective). Consequently, by [3] Lemma 3.8, the set of all h : H ⟶ G in Cov^{0}(G) which are locally definably isomorphic to v : V ⟶ G is cofinal in Cov^{0}(G) and hence the inverse limit p̃: G̃ ⟶ G is isomorphic to v : V ⟶ G. By Propositions 2.10 and 2.12 we obtain π(G) ≃ Kerv ≃ Aut(V/G) ≃ π_{1}(G) since π_{1}(V) = 1. Thus the result holds as required. □

Proof of Corollary 1.5: Let G be a definably t-connected definable group. By Proposition 3.4, π_{1}(G) is finitely generated and, by the isomorphism π_{1}(G) ≃ π(G) (Theorem 3.11) and [3] Proposition 3.11, π_{1}(G) is abelian. If G is abelian, then by [12] the assumptions of [3] Theorem 3.15 hold for G. Therefore we have π_{1}(G) ≃ π(G) ≃ ℤ^{l} and G[k] ≃ (ℤ/kℤ)^{l} for some l ∈ ℕ as required.

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