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Fundamental group in o-minimal structures with definable Skolem functions

In this paper we work in an arbitrary o-minimal structure with definable Skolem functions and we prove that definably connected, locally definable manifolds are uniformly definably path connected, have an admissible cover by definably simply connected, open definable subsets and, definable paths and definable homotopies on such locally definable manifolds can be lifted to locally definable covering maps. These properties allows us to obtain the main properties of the general o-minimal fundamental group, including: invariance and comparison results; existence of universal locally definable covering maps; monodromy equivalence for locally constant o-minimal sheaves - from which one obtains, as in algebraic topology, classification results for locally definable covering maps, o-minimal Hurewicz and Seifert - van Kampen theorems.


Introduction
In this paper we work in an arbitrary o-minimal structure M = (M, <, (c) c∈C , (f ) f ∈F , (R) R∈R ) which we assume to have definable Skolem functions. We are interested in developing algebraic topology tools for objects definable in M, more specifically a suitable fundamental group functor. One should point out here that, although the objects definable in M have a topology induced by the order in M, if M is non-archimedean, then all such objects (except the ones of dimension zero) are totally disconnected topological spaces and so the topological fundamental group is of no use.
In the case M expands a real closed field (M, <, 0, 1, +, ·), so this includes the semi-algebraic case, a suitable o-minimal fundamental group functor is already known and was studied in, for example, [7], [8], [3], [4], [18], [1] and [2], where its main properties were proved and applications to the theory of definable groups were obtained. These main properties include: (i) finite generation; (ii) invariance when going to o-minimal expansions of M or to bigger models of the first order theory of M and (iii) coincidence with the topological fundamental group when M = R. Often, in this case, the proofs of these properties relied on the o-minimal triangulation theorem ( [10]), a generalization of the semi-algebraic and of the semianalytic triangulation theorems ( [6], [25], [31] and [26]).
In the case M expands an ordered group (M, <, 0, +) a suitable o-minimal fundamental group functor was studied in the papers [21], [22], [12] and [13], where besides the properties mentioned above, the following were also proved: (iv) the existence of universal locally definable covering maps; (v) monodromy equivalence for locally constant o-minimal sheaves -from which one obtains, as in algebraic topology: (vi) classification results for locally definable covering maps; (vii) o-minimal Hurewicz and Seifert -van Kampen theorems.
In the case M is an arbitrary o-minimal structure with definable Skolem functions, following M. Mamino's ideas, a general o-minimal fundamental group was introduced in the paper [17], where Pillay's conjecture for definably compact groups was obtained in the general case. However, in that paper, the properties mentioned above were only proved for the o-minimal J-fundamental group which is the relativization of the general o-minimal fundamental group to a cartesian product The goal of this paper is to show, as conjectured in [17], all of the above properties for the general o-minimal fundamental group in the more general setting of o-minimal structures with definable Skolem functions.
The main technical result of the paper, obtained in Subsection 3.1, says that definably connected, locally definable manifolds are uniformly definably path connected, have an admissible cover by definably simply connected, open definable subsets and, definable paths and definable homotopies on such locally definable manifolds can be lifted to locally definable covering maps. See Properties 3.1. The results (i), (ii), (iii) and (iv) mentioned above are obtained, in the general case of M an arbitrary o-minimal structure with definable Skolem functions, in Subsection 3.2, and as explained in Subsection 3.1, they follow from Properties 3.1 in exactly the same way as in [13] for o-minimal expansions of ordered groups. In Subsection 3.2 we also show that if X is a definably connected locally definable manifold with definable charts in J, then the o-minimal J-fundamental group of X is isomorphic to the o-minimal fundamental group of X in M. Result (v) and its consequences (vi) and (vii) are mentioned in Subsection 3.3 after some background is recalled.
It might be possible that part of this theory may be developed in a general ominimal structure without the definable Skolem functions assumption. However, in the applications that we have in mind, all the structures have definable Skolem functions. It is even possible that under the assumption of the existence of definable Skolem functions some arguments may be simplified. Indeed, due to the trichotomy theorem it might be possible that there is a definable family of groups covering all the points of the domain of the structure. In that case the arguments in Subsection 3.1 would be closer to those of the o-minimal J fundamental group treated in [17]. We leave these issues for future work.
Regarding the applications, consider an algebraically closed valued field with a nontrivial valuation and consider the definable sets induced in the value group with a point at infinity added (the valuation of 0), Γ ∞ . Adding a copy of the value group Γ to Γ ∞ we obtain an o-minimal structure Σ with definable Skolem functions. Understanding the definable topology/definable algebraic topology of definable subsets of (Γ ∞ ) n can have applications in non-Archimedean tame topology by the main theorem of Hrushovski and Loeser ([27, Theorem 11.1.1]). For instance, Theorem 3.4 below is used in an essential way in developing the cohomology in the non-Archimedean tame setting [20]. Moreover, we expect that the main results about the fundamental group developed here or adaptations of those might give further applications.

Preliminaries
In this section we recall the notion of locally definable manifolds and locally definable covering maps and the general o-minimal fundamental group from [17].
Before we start, recall that an o-minimal structure M has definable Skolem functions if and only if for every uniformly definable family {X t } t∈T of nonempty definable subsets of some M k , there is a definable function h : T → M k such that: -h(t) ∈ X t for all t ∈ T.
2.1. Locally definable manifolds and covering maps. Here we recall the definition of the category of locally definable manifolds with continuous locally definable maps and the notion of locally definable covering maps.
A locally definable manifold S is equipped with the topology such that a subset U of S is open if and only if for each 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∈I0 U i and for each i ∈ I 0 , θ i (A ∩ U i ) is a definable subset of θ i (U i ). 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.
If U = {U α } α∈I is a cover of S by open locally 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 locally definable subsets, we say that A map f : X → Y between locally definable manifolds with definable charts (U i , θ i ) i≤κX and (V j , δ j ) j≤κY respectively is a locally definable map if for every finite I ⊆ κ X there is a finite J ⊆ κ Y such that: • 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, is a definable map between definable sets. Thus we have the category of locally definable manifolds with locally definable continuous maps. Definition 2.1. Given a 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 hold: A locally definable covering map p X : X → S is a locally definable covering map trivial over some admissible cover U = {U α } α∈I of S by open definable subsets.
We say that two locally definable covering maps p X : X → S and p Y : Y → S are locally definably homeomorphic if there is a locally definable homeomorphism F : X → Y such that: • p X = p Y • F. A locally definable covering map p X : X → S is trivial if it is locally definably homeomorphic to a locally definable covering map S × M → S : (s, m) → s for some set M.
Let p Y : Y → T be a locally definable covering map, X be a locally definable manifold and let f : . . , n, we define the d-interval, short for directed interval, I 1 ∧ · · · ∧ I n , whose domain is the set where c 1 , . . . , c n are n distinct points of M and ∼ is the equivalence relation defined by (c i , 1 Ii ) ∼ (c i+1 , 0 Ii+1 ) for each i = 1, . . . , n − 1 and identity elsewhere. The direction of I 1 ∧ · · · ∧ I n is 0 I1∧···∧In , 1 I1∧···∧In where 0 I1∧···∧In = c 1 , 0 I1 and 1 I1∧···∧In = c n , 1 In . The opposite of I 1 ∧ · · · ∧ I n is the d-interval (I 1 ∧ · · · ∧ I n ) op = [a 1 , b 1 ] ∧ · · · ∧ [a n , b n ], 0 (I1∧···∧In) op , 1 (I1∧···∧In) op with the same domain and opposite direction Ii are basic d-intervals, for i = 1, . . . , n, then (I 1 ∧ · · · ∧ I n ) op = I op n ∧ · · · ∧ I op 1 . Below, for the notion of definable space we refer the reader to [10, page 156]. We have: Due to Fact 2.4, below we will identify a d-interval I = I, 0 I , 1 I with its domain equipped with the definable total order < I . In particular, since the domain I of I op is a definable space of dimension one which is equipped with the definable total order > I , we have an order reversing definable homeomorphism (with respect to the topologies given by the orders) o I : I → I op given by the identity on the domain.
Given two d-intervals I = I 1 ∧ · · · ∧ I n and J = J 1 ∧ · · · ∧ J m , we define the d-interval I ∧ J = I 1 ∧ · · · ∧ I n ∧ J 1 ∧ · · · ∧ J m and we will regard I and J as definable subsets of I ∧ J . We say that I and J are equal, denoted I = J , if n = m and I i = J i for all i = 1, . . . , n.
Remark 2.5. Let I = I, 0 I , 1 I be a d-interval. If x, y ∈ I are such that x ≤ I y then the subset [x, y] I = {t ∈ I : x ≤ I t ≤ I y} of the elements of I between x and y is itself a d-interval. Indeed, let I i = [a i , b i ], 0 Ii , 1 Ii be basic d-intervals, for i = 1, . . . , n, and suppose that I = I 1 ∧ · · · ∧ I n . Let i x , i y ∈ {1, . . . , n} be such that Below, if X is a locally definable manifold and Y is a definable space, we say that h : Y → X is a definable continuous map if for some (equivalently, for every) definable subspace U of X with h(Y ) ⊆ U, the map h : Y → U is a definable continuous map between definable spaces. Definition 2.6. Let X be a locally definable manifold. A definable path α : I → X is a continuous (with respect to the topology on I given by the order) definable map from some d-interval I to X. We define α 0 := α(0 I ) and α 1 := α(1 I ) and call them the endpoints of the definable path α.
A definable path α : I → X is constant if α 0 = α(t) for all t ∈ I. Below, given a d-interval I and a point x ∈ X, we denote by c x I the constant definable path in X with endpoints x.
A definable path α : I → X is a definable loop if α 0 = α 1 . The inverse α −1 of a definable path α : I → X is the definable path A concatenation of two definable paths γ : I → X and δ : J → X with γ(1 I ) = δ(0 J ) is the definable path γ · δ : I ∧ J → X with: We say that X is definably path connected if for every u, v in X there is a definable path α : I → X such that α 0 = u and α 1 = v.
In the special case required for our applications we shall prove later, see Corollary 3.15 (1), that being definably connected is equivalent to being definably path connected.
Let X be a locally definable manifold and Y a definable space. Given two definable continuous maps f, g : Y → X, we say that a definable continuous map F (t, s) : Y × J → X is a definable homotopy between f and g if f = F 0 := F 0J and g = F 1 := F 1J , where F s := F (·, s) for all s ∈ J . In this case we say that f and g are definably homotopic, denoted f ∼ g.
Moreover, if γ : I → X is a definable path and J is any d-interval, then c γ0 I∧J · γ ∼ γ · c γ1 J ∧I . Since definable paths need not have the same domain, the notion of homotopic definable paths is not contained in the notion of homotopic definable maps just defined: Definition 2.8. Two definable paths γ : I → X, δ : J → X, with γ 0 = δ 0 and γ 1 = δ 1 , are called definably homotopic, denoted γ ≈ δ, if there are d-intervals I ′ and J ′ such that J ′ ∧ I = J ∧ I ′ , and there is a definable homotopy fixing the end points (i.e., they are definably homotopic by a definable homotopy F : K × A → X, where K = J ′ ∧ I = J ∧ I ′ , such that F (0 K , s) = γ 0 = δ 0 and F (1 K , s) = γ 1 = δ 1 for all s ∈ A.) We have: We also have: Proposition 2.13] Let X be a locally definable manifold and x 0 , x 1 ∈ X. Let P(X, x 0 , x 1 ) denote the set of all definable paths in X that start at x 0 and end at x 1 . Then the restriction of ≈ to P(X, By [17, Lemmas 2.14 e 2.15] we have: Definition 2.11. Let X be a locally definable manifold and e X ∈ X. If L(X, e X ) denotes the set of all definable loops that start and end at a fixed element e X of X (i.e. L(X, e X ) = P(X, e X , e X )). We define the o-minimal fundamental group π 1 (X, e X ) of X by and identity the class of a constant loop at e X .
If f : X → Y is a locally definable continuous map between two locally definable manifolds with e X ∈ X and e Y ∈ Y such that f (e X ) = e Y , then we have an induced homomorphism f * : Fact 2.12. [17, Corollary 2.18] Let X and Y be locally definable manifolds with e X ∈ X and e Y ∈ Y . Then (1) If X is definably path connected then π 1 (X, e X ) ≃ π 1 (X, x) for every x ∈ X.
As usual for a definably path connected locally definable manifold X if there is no need to mention a base point e X ∈ X, then by Fact 2.12 (1), we may denote π 1 (X, e X ) by π 1 (X).
Definition 2.13. Let X be a locally definable manifold. We define the o-minimal fundamental groupoid Π 1 (X) of X to be the small category Π 1 (X) given by ≈ We set [γ] := the class of γ ∈ P(X, x 0 , x 1 ). By Fact 2.10, the small category Π 1 (X) is indeed a groupoid with operations If f : X → Y is a locally definable continuous map between locally definable manifolds, then we have an induced functor f * : Π 1 (X) → Π 1 (Y ) which is a morphism of groupoids sending the object x ∈ X to the object f (x) ∈ Y and a morphism [γ]

Main results
3.1. The main properties. The goal in this subsection is to prove the following properties: Properties 3.1. Let P be the full subcategory of locally definable spaces in M whose objects are the locally definable manifolds. Then in the category P the following hold: (P1) (a) every object of P which is definably connected is uniformly definably path connected; (b) given a locally definable covering map p X : X → S in P then: (i) every definable path γ in S has a unique lifting γ which is a definable path in X with a given base point; (ii) every definable homotopy F between definable paths γ and σ in S has a unique lifting F which is a definable homotopy between the definable paths γ and σ in X. (P2) Every object of P has admissible covers by definably simply connected, open definable subsets refining any admissible cover by open definable subsets.
It follows, as observed in the concluding remarks (Section 5) of the paper [13], that with (P1) and (P2) above one proves in exactly the same way all the main results of the paper [13], now in the more general context of arbitrary o-minimal structures with definable Skolem functions. These results include all those mentioned in the Introduction.
In fact, besides (P1) and (P2) (and their consequences) everything else that is required is, on the one hand, results from [15], which hold in arbitrary o-minimal structures (and for locally definable spaces as well), and on the other hand, [10, Chapter 6, (3.6)], which is used to notice that the domains of the "good" definable paths are definably normal. In our case here the domains of the definable paths are Hausdorff, definably compact definable spaces (Fact 2.4), definable in the o-minimal structure M with definable Skolem functions and so they are definably normal by [16,Theorem 2.11].
The fact that (P1) and (P2) are the only requirements needed to develop the theory presented in [13] is somewhat natural. Indeed in topology, where we have good notions of paths and homotopies with the lifting of paths and homotopies property, all one needs is existence of such nice open covers as in (P2). In the ominimal context (here and in [13]), the role that (P1) (b) and (P2) play is similar to the role that the analogue properties play in topology. However, (P2) is often used in combination with the results from [15] mentioned above to get local definability. Also (P1) (a) is required essentially only once and to get local definability (see [13,Proposition 2.18]), the other places where it is used, it is used to replace definably connected by definably path connected.
Below let π : M n+1 → M n be the projection onto the first n coordinates and let τ : M n+1 → M : (x, y) → y be the projection onto the last coordinate.
We start by proving the existence of continuous definable sections for the projection of an open definable subset, over a finite cover by open definable subsets (Theorem 3.4 below). But first we recall a few facts.
The following is obtained from the definition of cells ([10, Chapter 3, §2]): Then by the definition of cells, C is a (i 1 , . . . , i n )-cell for some unique sequence (i 1 , . . . , i n ) of 0's and 1's. Moreover, if λ(1) < · · · < λ(d) are the indices λ ∈ {1, . . . , n} for which i λ = 1 and is the projection, then ). Then we clearly have Recall also the following fact: The result for a zero dimensional cell (d = 0) is immediate. For the inductive step, by Remark 3.2, after a reordering of coordinates, we may assume that our cell C is of the form - Then We still have to cover C \ W. If dim(C \ W ) < d then we apply the induction hypothesis and we are done. So suppose that dim( Observe that: Let V be the interior of ρ(C \ W ) and let Since U ⊆ K C\W ⊆ K C , we have that s ′ is continuous on U. We still have to cover (C \ W ) \ U. However since C ′ is open and dim ρ(C \ W ) = dim C ′ it follows that dim(ρ(C \ W ) \ V ) < d (see [10,Chapter 4, (1.9)]). So dim((C \ W ) \ U ) < d and we conclude by the induction hypothesis.
Let Z ⊆ M n+1 be a definable set, X a definable space and let f : X → π(Z) be a definable map. A lifting of f is a continuous map g : X → Z such that π•g = f .
Proof. Let µ : I → C be the definable path given by .
We will want to show that a given definable path is definably homotopic to a second definable path, but we cannot do it because their domains are not the same. For this we will modify the first definable path by patching to it appropriate constants that allow it to "wait" for the second definable path. That "waiting period" can occur either before or after (or both) appropriate basic d-intervals that composes the d-interval of the domain of the first path.
We say that a definable path γ ′ : I ′ → X is obtained from a definable path γ : I → X by modifying with constants if there are 0 I = t 0 < I t 1 < I · · · < I t r = 1 I and there are d-intervals J 0 , . . . , J r such that, if γ i := γ |[ti,ti+1]I for i = 0, 1, . . . r−1, then one of the following three cases holds: Lemma 3.8. If a definable path γ ′ : I ′ → X is obtained from a definable path γ : I → X by modifying with constants, then γ ≈ γ ′ .
Proof. By Theorem 3.4, there is a finite cover {U i : i = 1, . . . , m} of π(C) by open definable subsets such that for each i there is a continuous definable section s i : U i → C of the projection π (i.e. π • s i = id Ui ).
(2) Here we cannot just apply (1) to γ and σ since the liftings β and τ of the corresponding modifications by constants need not be definably homotopic.
Instead we need to use the definable homotopy F to build a sequence of definable paths which are "close" enough to guarantee that the liftings of the corresponding modifications by constants are definably homotopic. The sequence of definable paths will be obtained from a cell decomposition of the domain of the definable homotopy F compatible with the pull backs of the open definable subsets of π(C) on which we have continuous definable sections (Theorem 3.4) and the liftings of modifications by constants of these definable paths will be obtained using the continuous definable sections.
First assume that J is a basic d-interval [c, d], 0 J , 1 J . We may also assume that the definable total order < J on the domain [c, d] of J is < . If not, the argument is similar, one just has to construct the lifting from top to bottom instead of from bottom to top.
To proceed we also assume that I is a basic d-interval [a, b], 0 I , 1 I . We may furthermore assume that the definable total order < I on the domain [a, b] of I is < . If not the argument is similar, one just has to construct the lifting from right to left instead of from left to right.
Let  . For each two-dimensional cell C i,j and each l(i, j) such that C i,j ⊂ W l(i,j) , we have F (C i,j ) ⊂ U l(i,j) and for any twodimensional cells C i,j and C i ′ ,j ′ in [a, b]×[c, d], and for each l(i, j), l(i ′ , j ′ ), such that C i,j ⊂ W l(i,j) and Fix a sequence j = (j 0 , j 1 , . . . , , j r−1 ) with j i ∈ {0, 1, . . . , n i } for each i = 0, . . . , r − 1. For each i = 1, . . . , r let r j,i : ). This is the vertical path, going up on the right hand side of the "cell" ). This is the vertical path, going up on the left hand side of the "cell" be the definable path given by a j,i (t) = (t, f i,ji (t)).
a j,0 u j,1 a j,1 u j,2 a j,3 For k = 0, 1, . . . r − 1 let j[k] the sequence which is equal to j except in position k where it is min (j k + 1, n k ).
For k = r − 1 the argument is the same.
By Lemmas 3.7 and 3.8 and the transitivity of ∼ (Fact 2.7), to finish the proof it is enough to show that after modifying with constants, α j and α j[k] have liftings β j and β j[k] respectively such that ignoring modifications by constants.
For all the other cases the argument is similar. Now if I = I 1 ∧. . .∧I k with each I i a basic d-interval apply the previous process to each F |Ii×[c,d] , γ |Ii and σ |Ii to get γ ′i , σ ′i , β i and τ i such that β i ∼ τ i . If needed use (1) to replace the β i 's so that they patch together and similarly for the τ i 's. Patch all these to get γ ′ , σ ′ , β and τ and the result follows from transitivity and compatibility with concatenation of ∼ (Fact 2.7). Now if J = J 1 ∧ . . . ∧ J k with each J j a basic d-interval apply the previous process to each F |I×Jj and conclude by the transitivity of ∼ .
The main consequence of Lemma 3.10 is the following: Lemma 3.14. Let C ⊆ M n be a cell. Then: (1) C is definably path connected. In fact there is a uniformly definable family of definable paths connecting a given fixed point in C to any other point in C.
Proof. (1) The proof is by complete induction on the dimension n of the ambient space. If n = 0, the space is reduced to a single point and the result follows trivially. Assume that the result holds for every k < n. In order to show that it still holds for n we proceed by a new induction on the definition of cells. The zero-dimensional case is immediate. If C = Γ(f ) ⊆ M n+1 is the graph of a definable continuous function f : B → M , where B ⊆ M n is a cell, then the projection of C onto B is a definable homeomorphism and the result follows by the induction hypothesis. It remains to be considered the following case For each i fix u i ∈ U i . Since B is definably path connected, for each i, j, let γ i,j be a definable path in B such that γ i,j 0 = u i and γ i,j 1 = u j . If needed modify each γ i,j by constants and take by Lemma 3.10 (1) a lifting β i,j such that β i,j 0 = s i (u i ) and β i,j 1 = s j (u j ). Then there is a uniformly definable family of definable paths in C connecting any point of s i (U i ) to any point of s j (U j ). Since vertical paths are uniformly definable, the same holds for (π |C ) −1 (U i ) and (π |C ) −1 (U j ). Since C = (π |C ) −1 (U 1 ) ∪ . . . ∪ (π |C ) −1 (U m ) the result follows.
(2) The proof is again by complete induction on the dimension n of the ambient space. If n = 0, the space is reduced to a single point and the result follows trivially. Assume that the result holds for every k < n. In order to show that it still holds for n we proceed by a new induction on the definition of cells. The only case to be examined is the case Assume that C is an open set. Let α : I → C be a definable loop at p ∈ C. We want to show that α ≈ c p I . By the induction hypothesis, π • α ≈ c π(p) J . After modifying by constants, by Lemmas 3.8 and 3.10 (2) there are liftings β and τ of π • α and c π(p) J such that β ∼ τ. Since after modifying by constants, α and c p I are also liftings of π • α and c π(p) J , by Lemmas 3.7 and 3.8 and the transitivity of ∼ we obtain α ≈ c p I .
By Fact 3.3 and Lemma 3.14 we have the following which shows (P1) (a) and (P2). Compare with the corresponding results [13, Lemma 2.9 and Proposition 3.1] in o-minimal expansions of ordered groups.
Corollary 3.15. Let X be a definable manifold of dimension n. Then the following hold: (1) X is definably connected if and only if X is definably path connected. In fact, for any definably connected definable subset D of X there is a uniformly definable family of definable paths in D connecting a given fixed point in D to any other point in D. Lemma 3. 16. Let X and S be locally definable manifolds with definable charts. Suppose that p X : X → S is a locally definable covering map. Then the following hold.
(1) Let γ : I → S be a definable path in S. Let x ∈ X be such that p X (x) = γ 0 . Then there exists a unique definable path γ : I → X in X lifting γ such that γ 0 = x. (2) Suppose that F : I × J → S is a definable homotopy between the definable paths γ and σ in S. Let γ be a definable path in X lifting γ. Then there exists a definable path σ in X lifting σ and there exists is a unique definable lifting F : I × J → X of F , which is a definable homotopy between γ and σ.
Proof. Let U = {U α } α∈I be an admissible cover of S by open definable subsets over which p X : X → S is trivial. We may assume that U = {U α } α∈I refines the definable charts of S witnessing the fact that S is a locally definable manifold with definable charts.
(1) First we assume that I is a basic d-interval [a, b], 0 I , 1 I . We may also assume that the definable total order < I on the domain [a, b] of I is < . If not, the argument is similar, one just has to construct the lifting from right to left instead of from left to right.
Lift t1] , with γ 10 = x, using the definable is the definable connected component of p −1 (U l(0) ) in which x lays. Repeat the process for each γ i+1 = γ |[ti,ti+1] with γ i (t i ) instead of x. Patch the liftings together to obtain γ. Now if I = I 1 ∧. . .∧I k with each I i a basic d-interval apply the previous process to lift γ 1 = γ |I1 to γ 1 , with γ 10 = x and repeat the process for each γ i+1 = γ |Ii+1 with γ i (1 Ii ) instead of x. Patch the liftings together to obtain γ.
(2) First assume that J is a basic d-interval [c, d], 0 J , 1 J . We may also assume that the definable total order < J on the domain [c, d] of J is <. If not, the argument is similar, one just has to construct the lifting from top to bottom instead of from bottom to top.
To proceed we also assume that I is a basic d-interval [a, b], 0 I , 1 I . We may furthermore assume that the definable total order < I on the domain [a, b] of I is < . If not the argument is similar, one just has to construct the lifting from right to left instead of from left to right.
Let . For each two-dimensional cell C i,j and each l(i, j) such that C i,j ⊂ W l(i,j) , we have F (C i,j ) ⊂ U l(i,j) and for any two-dimensional cells C i,j and C i ′ ,j ′ in [a, b] × [c, d], and for each l(i, j), l(i ′ , j ′ ), such that C i,j ⊂ W l(i,j) and l(0,1) is the definable connected component of p −1 (U l(0,1) ) in which γ([t 0 , t 1 ]) lays. Repeat the process for each F 0,j+1 = F |C0,j+1 with F 0,j (Γ(f 0,j )) instead of γ([t 0 , t 1 ]). Patch the liftings together to ob- Now if I = I 1 ∧. . .∧I k with each I i a basic d-interval apply the previous process to lift F 1 = F |I1×[c,d] to F 1 , with F 1 (I 1 , c) = γ(I 1 ) and repeat the process for each F i+1 = F |Ii+1×[c,d] with γ(I i+1 ) instead of γ(I 1 ). Then patch these liftings together to obtain a definable lifting F : I × J → X of F which is a definable homotopy between γ and σ. Now if J = J 1 ∧ . . . ∧ J k with each J j a basic d-interval apply the previous process to lift F 1 = F |I×J1 to F 1 , with F 1 (I, 0 J1 ) = γ(I) and repeat the process for each F j+1 = F |I×Jj+1 with F j (I, 1 Jj ) instead of F 1 (I, 0 J1 ). To finish patch these liftings together to obtain a definable lifting F : I × [c, d] → X of F which is a definable homotopy between γ and σ.

3.2.
Universal covering maps and fundamental groups. As explained in Subsection 3.1, from the main properties of definable paths and definable homotopies (Properties 3.1) we obtain, in arbitrary o-minimal structures with definable Skolem functions, in exactly the same way as in [13] for o-minimal expansions of ordered groups, all of the results stated below.
(2) If X is Lindelöf, then the o-minimal fundamental group π 1 (X) of X is countable. In fact, if X is definable, then π 1 (X) is finitely generated.
For similar previously known results in special cases see [3], [2], [7], [12], [21] and [22]. (1) A universal locally J-definable covering map of X is J-definably homeomorphic to a universal locally definable covering map of X. Similarly, we have: Suppose that M is an o-minimal expansion of the ordered set of real numbers. Let X be a definably connected locally definable manifold. Then the following hold: (1) A topological universal covering map of X is topologically homeomorphic to the o-minimal universal locally definable covering map of X. (2) The topological fundamental group of X is isomorphic to the o-minimal fundamental group of X.
Remark 3.20. By Theorem 3.19 when M is an o-minimal expansion of the ordered set of real numbers, for definably connected locally definable manifolds, the theory developed in this paper coincides with the classical theory of topological covering maps ( [24]). However, one should point out that, in an arbitrary o-minimal structures M, the theory of topological covering maps is in some sense useless. In fact in that situation, if M is non-archimedean, then all definably connected locally definable manifolds are, with their natural topology, totally disconnected spaces and so have no non-trivial covering spaces. Our Theorem 3.17 shows that it is possible to find a suitable replacement of the theory of topological covering maps which in the archimedean case coincides with the classical theory and moreover it is preserved under elementary extensions (Theorem 3.18).
In the paper [17] it was convenient to introduced the o-minimal J-fundamental group which is the relativization of the general o-minimal fundamental group to a product of definable group-intervals. Our next goal is to show that these two kinds of o-minimal fundamental groups are isomorphic.
+∞}, together with a binary partial continuous definable operation + : J 2 → J and an element 0 ∈ J, such that: (i) x + y = y + x when defined; (x + y) + z = x + (y + z) when defined; if x < y and x + z and y + z are defined then x + z < y + z; (ii) for every x ∈ J, if x > 0, then the set {y ∈ J : x + y is defined} is an interval of the form (−b, r(x)); (iii) for every x ∈ J, we have lim z→0 (z + x) = x and if x > 0 we have also lim z→r(x) − (x + z) = b; (iv) for every x ∈ J there exists −x ∈ J such that x + (−x) = 0.
• If X is a locally definable manifold with definable J-charts, then a definable J-path (resp. constant definable J-path, or definable J-loop) is a definable path (resp. constant definable path or definable loop) α : I → X with I a d-J-interval; X is definably J-path connected if for every u, v in X there is a definable J-path α : I → X such that α 0 = u and α 1 = v. • If X and Y are locally definable manifolds with definable J-charts, then two definable continuous maps f, g : Y → X are definably J-homotopic, denoted f ∼ J g, if there is a definable homotopy F (t, s) : Y × J → X between f and g with J a d-J-interval; two definable J-paths γ : I → X, δ : J → X, with γ 0 = δ 0 and γ 1 = δ 1 , are definably J-homotopic, denoted γ ≈ J δ, if there are d-J-intervals I ′ and J ′ such that J ′ ∧ I = J ∧ I ′ , and there is a definable J-homotopy c γ0 J ′ · γ ∼ J δ · c δ1 I ′ fixing the end points.
The results mentioned in Subsection 2.2 for the relations ∼ and ≈ hold also for ∼ J and ≈ J respectively. Definition 3.24. Let X be a locally definable manifolds with definable J-charts, e X ∈ X and x 0 , x 1 ∈ X. Let P J (X, x 0 , x 1 ) denote the set of all definable J-paths in X that start at x 0 and end at x 1 and let L J (X, e X ) denotes the set of all definable J-loops that start and end at a fixed element e X of X (i.e. L J (X, e X ) = P J (X, e X , e X )). Then the restriction of ≈ J to P J (X, x 0 , x 1 ) × P J (X, x 0 , x 1 ) is an equivalence relation on P J (X, x 0 , x 1 ) and π J 1 (X, e X ) := L J (X, e X ) ≈ J is a group, the o-minimal J-fundamental group π J 1 (X, e X ) of X, with group operation given by [γ][δ] = [γ · δ] and identity the class a of constant J-loop at e X . Moreover, if f : X → Y is a locally definable continuous map between two locally definable manifolds with definable J-charts with e X ∈ X and e Y ∈ Y such that f (e X ) = e Y , then we have an induced homomorphism f * : π J 1 (X, e X ) → π J 1 (Y, e Y ) : [σ] → [f • σ] with the usual functorial properties.
As usual for a definably J-path connected locally definable manifold X with definable J-charts if there is no need to mention a base point e X ∈ X, then by Fact 2.12 (1), we may denote π J 1 (X, e X ) by π J 1 (X).
For J-definable paths and the o-minimal J-fundamental group we also have the corresponding properties (P1) and (P2). See [17,Corollary 3.21] for (P1) (a) and (P2) and see [17,Lemma 3.23] for (P1) (b). Thus, just like in Theorems 3.18 and 3.19, we can use these properties in the two setting to prove the following, which was conjectured in the paper [17]: Theorem 3.25. Let J = Π m i=1 J i be a cartesian product of definable group-intervals. Let X be a definably connected locally definable manifold with definable J-charts. Then the following hold: (1) there exists a universal locally definable covering map w : W → X where W is a locally definable manifold with definable J-charts. Moreover this locally definable covering is definably homeomorphic to a universal locally definable covering map of X. (2) The o-minimal J-fundamental group of X is isomorphic to the o-minimal fundamental group of X in M.
Proof. By properties (P1) and (P2) in J and the proof of [13, Theorem 1.2], X has a universal locally definable covering map w : W → X where W is a definably connected, locally definable manifold with definable J-charts, e W ∈ W and w(e W ) = e X . In particular, by [13, Remark 3.8], we have π J 1 (W, e W ) = 1. By Theorem 3.17, let u : U → X be a universal locally definable map with U definably connected, e U ∈ U and u(e U ) = e X . By [13,Remark 3.8], π 1 (U, e U ) = 1. Also there exists a locally definable covering map q : U → W such that is a commutative diagram and q(e U ) = e W .
Since X has definable J-charts, the same holds for U (we can refine the charts of U using the admissible cover given by (P2) in J). Therefore, u : U → X and q : U → W are also locally definable covering maps in J. Since π J 1 (U, e U ) ≃ q * (π J 1 (U, e U )) ≤ π J 1 (W, e W ) = 1 ([13, Corollary 2.17]), by [13,Remark 3.8], u : U → X is a universal locally definable covering map in J. Therefore, u : U → X and w : W → X are locally definably homeomorphic (actually in J) as required. Now note that the group Aut(U/X) of locally definable homeomorphisms φ : U → U such that u = u • φ, is the same in M and in J. By [13,Theorem 3.9] in M and in J respectively, we have π 1 (X, e X ) ≃ Aut(U/X) and π J 1 (X, e X ) ≃ Aut(U/X). Therefore, π 1 (X, e X ) ≃ π J 1 (X, e X ). 3.3. The monodromy. Recall that if X is a locally definable manifold, then X is equipped with the o-minimal site X def given by: (i) the category Op(X def ) of open definable subsets of X with morphisms being inclusions; (ii) the Grothendieck topology such that for U ∈ Op(X def ), a collection {U j } j∈J of objects of Op(X def ) is an admissible cover of U if it admits a finite subcover.
If C is any category admitting projective and inductive limits and satisfying the IPC property (see [30, Definition 3.1.10] for more details), then the category of C-pre-sheaves on the o-minimal site X def , denoted Psh C (X def ), is the category Fct(Op(X def ) op , C) of contravariant functors s → s |V from Op(X def ) to C with morphisms being natural transformations of such functors. The category of C-sheaves on the o-minimal site X def , denote Sh C (X def ), is the full subcategory of Psh C (X def ) whose objects satisfy the following gluing conditions: for every U ∈ Op(X def ) and every admissible cover {U j } j∈J of U we have: