Extensions of definable local homomorphisms in o-minimal structures and semialgebraic groups
aa r X i v : . [ m a t h . L O ] J a n EXTENSIONS OF DEFINABLE LOCAL HOMOMORPHISMS INO-MINIMAL STRUCTURES AND SEMIALGEBRAIC GROUPS
ELIANA BARRIGA
Abstract.
We state conditions for which a definable local homomorphism between twolocally definable groups G , G ′ can be uniquely extended when G is simply connected(Theorem 2.1). As an application of this result we obtain an easy proof of [3, Thm.9.1] (see Corollary 2.2). We also prove that Theorem 10.2 in [3] also holds for anydefinably connected definably compact semialgebraic group G not necessarily abelian overa sufficiently saturated real closed field R ; namely, that the o-minimal universal coveringgroup e G of G is an open locally definable subgroup of ^ H ( R ) for some R -algebraic group H (Thm. 3.3). Finally, for an abelian definably connected semialgebraic group G over R , we describe e G as a locally definable extension of subgroups of the o-minimal universalcovering groups of commutative R -algebraic groups (Theorem 3.4). Introduction and Preliminaries
The study of definable and locally definable groups has been of importance in the researchin model theory of o-minimal structures, and includes such classes as the semialgebraic andthe subanalytic groups. The ordered real field ( R , <, + , · ) as well as its expansion with theexponential function are examples of o-minimal structures [6]. Let M be a sufficiently κ -saturated o-minimal structure. By definable we mean definablein M . A group is called locally definable if the domain of the group and the graph of the groupoperation are a countable unions of definable sets. Every n -dimensional locally definablegroup G can be endowed with a unique topology τ making the group into a topologicalgroup such that any g ∈ G has a definable neighborhood definably isomorphic to M n [16,Prop. 2.2]. From now on, any topological property on a locally definable group refers tothis τ -topology, unless stated otherwise.When a locally definable group is definable, its τ -topology agrees with the t-topologygiven by Pillay in [18, Prop. 2.5]. The t-topology exists for every group G definable in ano-minimal structure N (not necessarily saturated), and when N o-minimally expands thereals, G is a real Lie group [18]. Mathematics Subject Classification.
Key words and phrases.
O-minimality, local homomorphisms, semialgebraic groups, real closed fields,algebraic groups, locally definable groups.Supported by the Israel Science Foundation (ISF) grants No. 181/16 and 1382/15.
A a locally definable subset X of a locally definable group G is τ - connected if X hasno nonempty proper definable subset ( τ -)clopen relative to X such that whose intersectionwith any definable subset of G is definable. When G is definable, by [18, Corollary 2.10],there is a unique maximal definably connected definable subset of G containing the identityelement of G , which we call the definable identity component of G , and we denoted it by G . Thus, G is definably connected if and only if G = G , or, equivalently by [18], if G hasno proper definable subgroup of finite index. We say that a definable group G is definablycompact if every definable path γ : (0 , → G has limits points in G (where the limits aretaken with respect to the t-topology on G ).The notions of path connectedness, homotopy, o-minimal fundamental group, and simplyconnectedness are defined as in algebraic topology using locally definable maps instead ofgeneral maps. For details on these definitions we refer the reader to [2].As in the Lie setting [5], Edmundo and Eleftheriou defined and proved in [9] the existenceof the o-minimal universal covering group e G of a connected locally definable group G . Asin Lie groups, e G covers any cover of G in this category, is simply connected, and allows tostudy definable and locally definable groups through them.In the theory of Lie groups, it is known that when two Lie groups are locally isomorphic,then their universal covers are (globally) isomorphic as Lie groups: Fact 1.1. ( [14, Corollary 4.20] ) Let H and H ′ be connected Lie groups, and e H Lie and f H ′ Lie their universal covering Lie groups respectively. Then H and H ′ are locally isomorphic ifand only if e H Lie and f H ′ Lie are isomorphic as Lie groups.
We say that two topological groups H and H ′ are locally homomorphic if there areneighborhoods U and U ′ of the identities of H and H ′ respectively and a map f : U ⊆ H → U ′ ⊆ H ′ such that f ( hh ′ ) = f ( h ) f ( h ′ ) whenever h, h ′ , and hh ′ belong to U [5, Definition2, Chap. 2, Section 7]; such a map f is called a local homomorphism of H into H ′ , and if inaddition f is a homeomorphism, f is called a local isomorphism , and H and H ′ are locallyisomorphic .From a model-theoretical point of view, it is natural to ask whether Fact 1.1 holds ornot in the category of locally definable groups.For this, we will restrict the previous definitions to definable maps. However, if M expands an ordered field ( R, <, + , , · , , the additive group ( R, +) is definably locallyisomorphic to the group G = ([0 , , + mod ) with addition modulo through the map f : ( − , ) ⊆ ( R, +) → G , f ( t ) = t mod , but ( R, +) and e G = (cid:0)S n ∈ N ( − n, n ) , + (cid:1) ≤ ( R, +) are not isomorphic as locally definable groups ( ( R, +) is definable, but e G is not).Fact 1.1 follows from a well-known result for topological groups that assures that a localhomomorphism between topological groups with domain a connected neighborhood of theidentity element of a simply connected group H can be extended to a group homomorphismfrom the whole group H : Fact 1.2. ( [5, Thm. 3, Chap. 2, Section 7] ) Let H be a simply connected topological group.Let f be a local homomorphism of H into a group H ′ . If the set on which f is defined isconnected, then it is possible to extend f to a homomorphism f : H → H ′ . XTENSIONS OF DEFINABLE LOCAL HOMOMORPHISMS IN O-MINIMAL STRUCTURES AND SEMIALGEBRAIC GROUPS3
Again the above example of ( R, +) and G = ([0 , , + mod ) with the definable localisomorphism f : ( − , ) ⊆ ( R, +) → G , f ( t ) = t mod shows that f cannot be extendedto a locally definable homomorphism from ( R, +) into G (otherwise, the kernel, Z , ofsuch a (locally) definable homomorphism would be definable in M , which is not possible).Therefore, Fact 1.2 does not hold in the category of locally definable groups.Nevertheless, we were able to formulate a sufficient condition for a definable local ho-momorphism to extend to a locally definable homomorphism of the whole group. Moreprecisely, we prove the following. Theorem 2.1.
Let G and G ′ be locally definable groups, U a definably connected definableneighborhood of the identity e of G , and f : U ⊆ G → G ′ a definable local homomorphism.Assume that there is a definable neighborhood V of e generic in G such that V − V ⊆ U . If G is simply connected, then f is uniquely extendable to a locally definable group homomorphism f : G → G ′ . Above a subset X of a group G , locally definable in a κ -saturated o-minimal structure,is left (right) generic in G if less than κ -many left (right) group translates of X cover G ;i.e., G = AX ( G = XA ) for some A ⊆ G with | A | < κ . X is generic if it is both left andright generic. A locally definable group may not have definable generic subsets; however,when it does, the group has interesting properties, see for example [10, Thm. 3.9].Theorem 2.1 allows us to easily prove Corollary 2.2, a result on extension of a defin-able local homomorphism between abelian locally definable groups that we have previouslyproved in [3, Thm. 9.1] using different methods. From now on until the end of this paper, let R = ( R, <, + , · ) be a sufficiently saturatedreal closed field, and denote by R a its additive group ( R. +) and by R m its multiplicativegroup of positive elements (cid:0) R > , · (cid:1) . Corollary 2.2 was applied in [3] to prove a characterization of the o-minimal universalcovering group e G of an abelian definably connected definably compact semialgebraic group G over R in terms of R -algebraic groups [3, Thm. 10.2] (see Fact 3.1).In Section 3.1, we show that Theorem 10.2 in [3] also holds for any definably connecteddefinably compact semialgebraic group over R not necessarily abelian (Thm. 3.3). Theorem 3.3.
Let G be a definably connected definably compact group definable in R .Then e G is an open locally definable subgroup of ^ H ( R ) for some R -algebraic group H . By [17], every abelian torsion free semialgebraic group over R is definably isomorphic to R ka × R nm for some k, n ∈ N . Therefore, the results for torsion free and definably compactsemialgebraic groups over R suggest asking ourselves the following. Question 1.3.
Let G be an abelian definably connected semialgebraic group over R . Is e G an open locally definable subgroup of ^ H ( R ) for some R -algebraic group H ?Although the above question remains open, we were able to prove: Theorem 3.4.
Let G be a definably connected semialgebraic group over R . Then thereexist a locally definable group W , commutative R -algebraic groups H , H such that e G is a ELIANA BARRIGA locally definable extension of W by H ( R ) where W is an open subgroup of ^ H ( R ) . Infact, H ( R ) is isomorphic to R ka × R nm as definable groups for some k, n ∈ N . Where we say that a locally definable group G is a locally definable extension of G ′ by G ′′ if we have an exact sequence → G ′′ → G → G ′ → in the category of locally definablegroups with locally definable homomorphisms [8, Section 4].2. An extension of a definable local homomorphism between locallydefinable groups
Recall that we are working in a sufficiently κ -saturated o-minimal structure M . Theorem 2.1.
Let G and G ′ be locally definable groups, U a definably connected definableneighborhood of the identity e of G , and f : U ⊆ G → G ′ a definable local homomorphism.Assume that there is a definable neighborhood V of e , generic in G , such that V − V ⊆ U . If G is simply connected, then f is uniquely extendable to a locally definable grouphomomorphism f : G → G ′ .Proof. Let x ∈ G . Since G is path connected, there is a locally definable path ω : I =[0 , → G such that ω (0) = e , ω (1) = x . Note that since V is generic in G , then thetopological interior of V is also generic in G – this fact might be followed from Lemma 2.22of [13], for example – . So replacing V with its topological interior, we now have that V isan open neighborhood of e and generic in G . By the genericity of V in G , G = A · V forsome A ⊆ G , | A | < κ . As ω ( I ) is a definable subset of A · V , saturation yields that thereis a finite set A ⊆ A such that ω ( I ) ⊆ A · V . Then, I = S a ∈ A ω − ( a · V ) . As V is anopen neighborhood, ω − ( a · V ) is also an open neighborhood of some element of I .Since M is an o-minimal structure, ω − ( a · V ) is a finite union of points and intervals.Then (cid:8) ω − ( a · V ) : a ∈ A (cid:9) is a collection of open intervals in I . Thus we can choosea division of I into subintervals [ t i , t i +1 ] such that t < t < . . . < t n = 1 and ( ω [ t i , t i +1 ]) ⊆ a i V for some a i ∈ A . So, for t, t ′ ∈ [ t i , t i +1 ] , ω ( t ) = a i v , ω ( t ′ ) = a i v ′ for v, v ′ ∈ V , and ω ( t ) − ω ( t ′ ) = v − v ′ ∈ V − V ⊆ U .For such a locally definable path ω and division define f ω ( x ) := f (cid:16) ω ( t ) − ω ( t ) (cid:17) f (cid:16) ω ( t ) − ω ( t ) (cid:17) · · · f (cid:16) ω ( t n − ) − ω ( t n ) (cid:17) . Now, we will show that f ω is invariant under refinements of the division of I .Let t ′ ∈ [ t i , t i +1 ] be a new subdivision point. Since ω ( t i ) − ω ( t ′ ) , ω ( t ′ ) − ω ( t i +1 ) ∈ U and f is a local homomorphism, f (cid:16) ω ( t i ) − ω ( t ′ ) (cid:17) f (cid:16) ω ( t ′ ) − ω ( t i +1 ) (cid:17) = f (cid:16) ω ( t i ) − ω ( t i +1 ) (cid:17) .Hence, given two subdivisions of I , we can consider a refinement common to these, then f ω does not depend on the subdivisions of I .Now, we will show that f ω is determined independently of the choice of a path ω . Let ω ′ : I → G be another locally definable path connecting e and x . Since G is simplyconnected, there is a locally definable homotopy Γ : I × I → G between ω and ω ′ with Γ ( t,
0) = ω ( t ) and Γ ( t,
1) = ω ′ ( t ) . As Γ ( I × I ) is a definable subset of A · V , again XTENSIONS OF DEFINABLE LOCAL HOMOMORPHISMS IN O-MINIMAL STRUCTURES AND SEMIALGEBRAIC GROUPS5 saturation implies that there is a finite set A ⊆ A such that Γ ( I × I ) ⊆ A · V . So, I × I ⊆ S a ∈ A Γ − ( a · V ) . By continuity of Γ , for every ( t, t ′ ) ∈ I × I , there is I i × I j ⊆ I × I such that ( t, t ′ ) ∈ I i × I j , Γ ( I i × I j ) ⊆ a i,j V for some a i,j ∈ A . Therefore, we can partition I in a finite number of subintervals I i such that Γ ( I i × I j ) ⊆ a i,j V for some a i,j ∈ A . Asthe subintervals I i are finite and cover I , we may assume that I i = (cid:2) in , i +1 n (cid:3) for some n ∈ N .Note that if ( s, t ) , ( s ′ , t ′ ) ∈ I i × I j , then Γ ( s, t ) − Γ ( s ′ , t ′ ) = ( a i,j v ) − a i,j v ′ = v − v ′ ∈ U .Let ω i ( t ) := Γ (cid:0) t, in (cid:1) for i ∈ { , , . . . , n } . So ω ( t ) = ω ( t ) , ω n ( t ) = ω ′ ( t ) . Since f is alocal homomorphism, then f ω i ( x ) = f ω i +1 ( x ) for i ∈ { , , . . . , n − } . Therefore, f ω isdetermined independently of the choice of a path ω , and denote it by f .Now, let x, y ∈ G and ω, γ : I → G locally definable paths connecting e and x , and e and y ,respectively. Then the locally definable path σ : I → G , σ ( t ) := xγ ( t ) connects x with xy .Let ω ∗ σ denote the concatenation of the paths ω and σ . Then, f ω ( x ) f γ ( y ) = f ω ∗ σ ( xy ) ,namely f ( x ) f ( y ) = f ( xy ) , so f is a group homomorphism.Next, we will see that f is an extension of f . As U is definably connected, so is pathconnected [2], then if x ∈ U , there is a locally definable path ω : I → G such that ω (0) = e , ω (1) = x , and ω ( I ) ⊆ U . As f does not depend on the subdivisions of I , let t = 0 , t = 1 ,then it is clear that f ( x ) = f (cid:16) ω ( t ) − ω ( t ) (cid:17) = f ( x ) .Now, let h be another extension of f . Let G = (cid:8) g ∈ G : h ( g ) = f ( g ) (cid:9) . Then G is alocally definable subgroup of G , and U ⊆ G . As U is generic in G , U generates G [10, Fact2.3(2)], then G ⊆ G , so h = f . Therefore, f is uniquely extendable to a locally definablegroup homomorphism from G into G ′ .Finally, observe that f is a locally definable map on G . For this note that since f is agroup homomorphism, f (cid:0) x − y (cid:1) = f ( x ) − f ( y ) = f ( x ) − f ( y ) for every x, y ∈ U , then f restricted to Q n U − U is a definable map. And as G = h U i = S n ∈ N Q n U − U , then f is alocally definable map on G . (cid:3) For a locally definable group G in M , we denote by G the smallest, if such exists, type-definable subgroup of G of index smaller than κ , where for a small set we mean a subsetof M n with cardinality smaller than κ ([11]). G may not exist (see an example in [10,Subsection 2.2]). For definable groups such a type-definable group always exists [19].With Theorem 2.1, it is easy to prove the following result, which was previously demon-strated in [3] using a different technique. Corollary 2.2. ( [3, Thm. 9.1] ) Let G and G ′ be two abelian locally definable groups suchthat G is connected, torsion free, and G exists. Let U ⊆ G be a definably connecteddefinable set such that G ⊆ U , and f : U ⊆ G → G ′ a definable local homomorphism.Then f is uniquely extendable to a locally definable group homomorphism f : G → G ′ .Proof. We will just check that the assumptions are enough to apply Theorem 2.1. First,we will see that G is simply connected. Since G exists, then G is definably generated, andby [10, Thm 3.9], G covers an abelian definable group. Claim 6.4 in [3] yields that G issimply connected since G is also torsion free. Now, as G ⊆ U , by saturation, there is a ELIANA BARRIGA definable set V such that G ⊆ V ⊆ V − V ⊆ U . Let V be the topological interior of V in G , which is definable, then G ⊆ V ⊆ V − V ⊆ U since G is open in G . Therefore, V is a definable neighborhood of the identity in G open in G , generic in G , and V − V ⊆ U .Finally, Theorem 2.1 implies that there is a unique extension f : G → G ′ f that is a locallydefinable group homomorphism. (cid:3) Universal covers of definably local homomorphic locally definablegroups
As in [3], Corollary 2.2, together with other results in [3], implies the following fact,and establishes a relation between the o-minimal universal covering groups of two defin-ably locally homomorphic locally definable groups. This result could be interpreted as ananalogue in the category of locally definable groups of the known fact that two connectedlocally homomorphic Lie groups have isomorphic universal covering Lie groups (Fact 1.1).
Fact 3.1. ( [3, Thm. 10.1] ) Let G and G ′ be two divisible abelian connected locally definablegroups such that G exists and G is a decreasing intersection of ω -many simply connecteddefinable subsets of G .Let X ⊆ G be a definable set with G ⊆ X , and f : X ⊆ G → G ′ a definable homeomor-phism and local homomorphism. Then e G is an open locally definable subgroup of e G ′ . An important corollary of the above result is the characterization of the o-minimal uni-versal covers of the abelian definably connected definably compact semialgebraic groupsover R in terms of algebraic groups. Fact 3.2. ( [3, Thm. 10.2] ) Let G be an abelian definably connected definably compactgroup definable in R . Then e G is an open locally definable subgroup of ^ H ( R ) for someZariski-connected R -algebraic group H . In fact, the algebraic group H in Fact 3.2 is commutative since G is abelian, and byTheorem 4.1 in [3], G and H ( R ) are definably locally homomorphic.3.1. Universal cover of a definably compact semialgebraic group.
Now, we willshow that Fact 3.2 also holds for every definably connected definably compact R -definablegroup not necessarily abelian. For this, we will use several results of Hrushovski, Peterzil,and Pillay in [12] together with the abelian case Fact 3.2. Theorem 3.3.
Let G be a definably connected definably compact group definable in R .Then e G is an open locally definable subgroup of ^ H ( R ) for some R -algebraic group H .Proof. By [12, Corollary 6.4], G is definably isomorphic to the almost direct product of S and G where S is some definably connected semisimple definable group, and G is someabelian definably connected definably compact group, so G ≃ ( G × S ) /F for some finitecentral subgroup F ⊆ G × S . Therefore, G × S is a finite cover of G , then e G ≃ ^ G × S .Since the o-minimal fundamental group for locally definable groups (see [2] ) has the prop-erty that the group π ( G , g ) × π ( G , g ) is isomorphic to the group π ( G × G , ( g , g )) XTENSIONS OF DEFINABLE LOCAL HOMOMORPHISMS IN O-MINIMAL STRUCTURES AND SEMIALGEBRAIC GROUPS7 for G , G locally definable groups and g , g elements in G , G , respectively, then if G and G are simply connected, then ^ G × G is isomorphic to f G × f G as locally definable groups.Hence, ^ G × S ≃ f G × e S .Now, by [12, Fact 1.2(3)], the center Z ( S ) of S is finite and S/Z ( S ) is definably iso-morphic to a direct product S × . . . × S n of finitely many definably simple groups S i ’s.By [12, Fact 1.2(1)], S i ≃ H i ( R i ) for some real closed field R i definable in R and some R i -algebraic group H . But, by [15, Thm. 1.1], R i ≃ R . Let H ⋆ := H × . . . × H n , then H ⋆ ( R ) ≃ H ( R ) × . . . × H n ( R ) ≃ S × . . . × S n ; namely, every semisimple semialge-braic group S over R is up to its center H ⋆ ( R ) for some R -algebraic group H ⋆ . Thus, e S ≃ ^ H ⋆ ( R ) .Now, by Fact 3.2, f G ≤ ^ H ⋆⋆ ( R ) for some R -algebraic group H ⋆⋆ , then e G ≃ f G × e S ≤ ^ H ⋆⋆ ( R ) × ^ H ⋆ ( R ) ≃ ^ H ( R ) for H := H ⋆⋆ × H ⋆ . Note that e G and ^ H ( R ) have thesame dimension, then e G is, up to isomorphism of locally definable groups, an open locallydefinable subgroup of ^ H ( R ) for some R -algebraic group H . (cid:3) Universal cover of an abelian semialgebraic group.
In this subsection, we willprove that the o-minimal universal covering group of an abelian definably connected semi-algebraic group over R is a locally definable (group) extension, in the category of locallydefinable groups (see [8, Section 4] for basics on locally definable extensions), of an openlocally definable subgroup of ^ H ( R ) by H ( R ) for some R -algebraic groups H , H .This will mainly follow by the characterization of abelian groups definable in o-minimalstructures [7] and our previous results in this work.Recall that for the sufficiently saturated real closed field R = ( R, <, + , · ) , R a denotes theadditive group ( R. +) , and R m the multiplicative group of positive elements (cid:0) R > , · (cid:1) . Theorem 3.4.
Let G be a definably connected semialgebraic group over R . Then thereexist a locally definable group W , commutative R -algebraic groups H , H such that e G is alocally definable extension of W by H ( R ) where W is an open subgroup of ^ H ( R ) . Infact, H ( R ) is isomorphic to R ka × R nm as definable groups for some k, n ∈ N .Proof. By [7], G is a definable extension of some abelian definably compact definably con-nected definable group K by the maximal torsion free normal definable subgroup T of G : → T → G → K → . By [3, Thm. 10.2], e K ≤ ^ H ( R ) for some commutative R -algebraicgroup H .By [17], T is definably isomorphic to R ka × R nm for some k, n ∈ N . So in particular, T ≃ H ( R ) for H = (cid:0) R (cid:0) √− (cid:1) , + (cid:1) k × (cid:0) R (cid:0) √− (cid:1) , · (cid:1) n . Thus, so far we have that → H ( R ) → G π → K → with e K ≤ ^ H ( R ) for some commutative R -algebraic groups H , H . ELIANA BARRIGA
By [17, Thm. 5.1], there is a continuous definable section s : K → G (continuous withrespect with their τ -topologies). Then the map ϕ : H ( R ) × K → G , ϕ ( h, k ) = hs ( k ) is a definable homeomorphism with inverse ϕ − ( g ) = (cid:16) g ( s ( π ( g ))) − , π ( g ) (cid:17) for g ∈ G .Here the direct product H ( R ) × K has the product topology, and the groups K , G , andthe subgroup H ( R ) ≤ G have the τ -topology ([16]) which coincides with the t-topology([18]) for definable groups.Let f be the definable two-cocycle associated with the section s , i.e., f : K × K → H ( R ) ( k , k ) s ( k ) s ( k ) ( s ( k k )) − . Then, G is definably isomorphic to the group (cid:16) H ( R ) × K, · f (cid:17) with group operationgiven by ( h, k ) · f (cid:0) h ′ , k ′ (cid:1) = (cid:0) hh ′ f (cid:0) k, k ′ (cid:1) , kk ′ (cid:1) , through the definable group isomorphism ϕ .Let p K : e K → K be the o-minimal universal covering homomorphism of K , and id : H ( R ) → H ( R ) the identity map on H ( R ) .Now, let e f : e K × e K → H ( R ) (cid:16) e k , e k (cid:17) f (cid:16) p K (cid:16) e k (cid:17) , p K (cid:16) e k (cid:17)(cid:17) . The two-cocycle condition ([7, Eq. 3, Section 3]) of f implies the same condition for e f ,thus the group (cid:16) H ( R ) × e K, · e f (cid:17) with group operation given by (cid:16) h, e k (cid:17) · e f (cid:16) h ′ , e k (cid:17) = (cid:16) hh ′ e f (cid:16) e k , e k (cid:17) , e k e k (cid:17) induced by e f is a locally definable group. Let i : H ( R ) → H ( R ) × e K be the map h ( h, , and π : H ( R ) × e K → e K the projection map into the second coordinate. Sofar we have that → H ( R ) i → (cid:16) H ( R ) × e K, · e f (cid:17) π → e K → is a locally definable extension. Note that the locally definable group (cid:16) H ( R ) × e K, · e f (cid:17) isconnected because H ( R ) and e K are both connected [8, Corollary 4.8(ii)].Now, the map ϕ ◦ ( id × p K ) : (cid:16) H ( R ) × e K, · e f (cid:17) → G (cid:16) h, e k (cid:17) ϕ (cid:16) h, p K (cid:16)e k (cid:17)(cid:17) = hs (cid:16) p K (cid:16)e k (cid:17)(cid:17) XTENSIONS OF DEFINABLE LOCAL HOMOMORPHISMS IN O-MINIMAL STRUCTURES AND SEMIALGEBRAIC GROUPS9 is a locally definable covering map. The abelianity of G and the definition of e f let easilyto conclude that ϕ ◦ ( id × p K ) is also a group homomorphism. Hence, ϕ ◦ ( id × p K ) : (cid:16) H ( R ) × e K, · e f (cid:17) → G is a locally definable covering homomorphism.Next, we will see that (cid:16) H ( R ) × e K, · e f (cid:17) is simply connected. Note that, by [4, Propo-sition 5.14], e K is torsion free. So H ( R ) and e K are both torsion free, and therefore (cid:16) H ( R ) × e K, · e f (cid:17) is too. Finally, [3, Claim 6.4] yields the simply connectedness of thegroup (cid:16) H ( R ) × e K, · e f (cid:17) , then ϕ ◦ ( id × p K ) : (cid:16) H ( R ) × e K, · e f (cid:17) → G is the o-minimal universal covering homomorphism of G . We conclude the proof of thetheorem. (cid:3) Acknowledgements
I warmly thank Assaf Hasson and Kobi Peterzil for their support, generous ideas, andkindness during this work. I also want to express my gratitude to the Ben-Gurion Universityof the Negev, Israel joint with its warm and helpful staff for supporting my research.The main results of this paper have been presented at the Israel Mathematical Union(IMU) virtual meeting 2020 on September 6th, 2020 (Israel), the Logic virtual seminar ofthe Università degli Studi della Campania “Luigi Vanvitelli” on May 28th, 2020 (Campania,Italy), and the Logic seminar of the Hebrew University of Jerusalem (Jerusalem, Israel) onDecember 11th, 2019.This research was funded by the Israel Science Foundation (ISF) grants No. 181/16 and1382/15.
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Eliana Barriga, Department of Mathematics, Ben Gurion University of the Negev, Be’er-Sheva 84105, Israel
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