Molino's description and foliated homogeneity

MOLINO’S DESCRIPTION AND FOLIATED HOMOGENEITY JES´ US A. ´ ALVAREZ L´ OPEZ AND RAM´ ON BARRAL LIJ´ O Abstract. The topological Molino’s description of equicontinuous foliated spaces, studied by the first author and Moreira Galicia, gives conditions to reduce their study to the particular case where the holonomy pseudogroup can be represented by a pseudogroup on some local group Ggenerated by some of its local left translations (a G-foliated space). That description is sharpened in this paper by introducing a foliated action of a compact topological group on the resulting G-foliated space, like in the case of Riemannian foliations. Moreover a C∞version is also studied. The triviality of this compact group characterizes compact minimal G-foliated spaces, which are also characterized by their foliated homogeneity in the C∞case. We also give an example where the projection of the Molino’s description is not a principal bundle, and another example of positive topological codimension where the foliated homogeneity cannot be checked by only comparing pairs of leaves—in the case of zero topological codimension, weak solenoids with this property were given by Fokkink and Oversteegen, and later by Dyer, Hurder and Lukina. Contents 1. Introduction 2 2. Preliminaries 5 3. Molino’s description 19 4. Foliated homogeneous foliated spaces 24 5. C∞Molino’s description 26 6. Right local transverse actions 27 7. C∞G-foliated spaces are C∞foliated homogeneous 32 8. Examples and open problems 32 References 37 Date: February 14, 2019. 1991 Mathematics Subject Classification. 57R30. Key words and phrases. Foliated space, equicontinuous, strongly quasi-analytic, Molino’s description, foliated homogeneous. The authors are partially supported by MICINN, grant MTM2014-56950-P, and Xunta de Galicia, grant 2015 GPC GI-1574. 1 2 J.A. ´ ALVAREZ L ´ OPEZ AND R. BARRAL LIJ ´ O 1. Introduction A description of certain compact minimal equicontinuous foliated spaces was given by the first author and Moreira Galicia [10]. It can be considered as a topological version of the Molino’s description of Riemannian foliations on compact manifolds [37], in the minimal case. This gave another confirmation that equicontinuous foliated spaces should be considered as the topological Riemannian foliations, as asserted by Ghys [37, Appendix E]. That description reduces the study of such foliated spaces to the particular case of G-foliated spaces, which are the foliated spaces whose holonomy pseudogroup can be represented by a pseudogroup on some local group G generated by some of its local left translations. The classical example of G-foliated spaces are Lie foliations, which are used in the original Molino’s theory to describe Riemannian foliations. According to the role played by Molino’s theory in the study of Riemannian foliations, its topological version should have interesting applications; for instance, it was already used in [10] to study the growth of the leaves. Dyer, Hurder and Lukina also gave an analogue of Molino’s description for equicontinuous matchbox manifolds [18, 19], which is the case of compact connected minimal foliated spaces of topological codimension zero; i.e., with totally disconnected local transversals. The advantage of their construction is that it works without any additional condition, but their description is unique just when our hypotheses are fulfilled. Our first goal is to show the following slight sharpening of the main result of the topological Molino’s theory (Section 3). The terminology and notation used here are recalled in Section 2. Theorem A (Cf. [10, Theorem A]).Suppose that a foliated space X≡ (X, F)is compact, minimal, equicontinuous and strongly quasi-analytic, and the closure of its holonomy pseudogroup is also strongly quasi-analytic. Then there is a local group G, a compact topological group H, a compact minimal G-foliated space  X0≡( X0, F0), a foliated map ˆπ0∶ X0→X, and a free foliated right H-action on  X0such that the restrictions of ˆπ0to the leaves of  X0are the holonomy coverings of the leaves of X, and ˆπ0induces a homeomorphism  X0/H→X. Precisely, our new contribution in Theorem A is the existence of Hsatisfying the stated properties. If His the representative of the holonomy pseudogroup of Xon a space Tinduced by the choice of a good foliated atlas, and we fix some u0∈T, then His the group of germs at u0of the maps gin the closure Hwith u0∈domgand g(u0)=u0. Following the construction of  X0in [10], we get a compatible compact topology on Hand a right foliated H-action on  X0satisfying the statement of Theorem A. We also show that the construction of (G, H,  X0,ˆπ0)is independent of the choices involved up to an obvious equivalence relation (Proposition 3.1), and therefore (G, H,  X0,ˆπ0)is called the Molino’s description of X; in particular, MOLINO’S DESCRIPTION AND FOLIATED HOMOGENEITY 3 Gis called the structural local group according to [37, 10], and His called the discriminant group according to [18]. Under the hypothesis of Theorem A, we also prove the following additional properties: ●Xis a G-foliated space for some local group Gif and only if its discriminant group is trivial (Proposition 3.2). ●There is a subgroup in Hisomorphic to the holonomy group of every leaf (Proposition 3.4). ●If Xis C∞, then its Molino’s description becomes C∞in a unique obvious sense (Proposition 5.1). ●The map ˆπ0may not be a fiber bundle (an example is given in Section 8.2). This is the only missing property when comparing with the Riemannian foliation case. Our second goal is to characterize G-foliated spaces using a property called foliated homogeneity. A foliated space X≡(X, F)is called foliated homogeneous if the group Homeo(X, F)of its foliated transformations acts transitively on itself (a foliated version of homogeneity). This notion was studied by Clark and Hurder in the case of matchbox manifolds [15], where homogeneity and foliated homogeneity are equivalent notions because Homeo(X)=Homeo(X, F)since the leaves are the path connected components. Clark and Hurder have shown that a matchbox manifold is equicontinuous if and only if it is a weak solenoid (an inverse limit of a tower of covering maps between closed connected manifolds), and it is homogeneous if and only if it is a McCord solenoid (the covering maps can be chosen to be regular), also called strong solenoid. Since McCord solenoids are transversely modeled by left translations on profinite groups, they are particular cases of G-foliated spaces. For this reason, the mentioned Molino’s description of Dyer, Hurder and Lukina is a procedure to construct McCord solenoids from weak solenoids. On the other hand, according to the original Molino’s theory [37], among minimal Riemannian foliations on closed manifolds, the homogeneous ones are the Lie foliations (the G-foliations for Lie groups G). Thus, generalizing the case of matchbox manifolds and minimal Riemannian foliations on closed manifolds, it makes sense to ask whether any compact minimal foliated space is foliated homogeneous if and only if it is a G-foliated space. We give the following answers. Theorem B. If a foliated space Xis compact, minimal and foliated homogeneous, then it satisfies hypotheses of Theorem A and is a G-foliated space for some local group G. Theorem C. Suppose that a foliated space Xis compact, minimal and C∞. Then the following conditions are equivalent: (i) Xis C∞foliated homogeneous. (ii) Xis foliated homogeneous. (iii) Xsatisfies the hypotheses of Theorem A and is a G-foliated space for some local group G. 4 J.A. ´ ALVAREZ L ´ OPEZ AND R. BARRAL LIJ ´ O Here, a foliated space is said to be C∞when it has a foliated atlas whose changes of coordinates are C∞along the leaves, and their leafwise partial derivatives of arbitrary order are continuous (on the ambient space). Other related concepts are defined in the same way, like C∞foliated maps, C∞ diffeomorphisms, (leafwise) tangent space, (leafwise) Riemannian metrics, (leafwise) Riemannian foliated spaces, etc. For C∞foliated spaces, the concept of C∞foliated homogeneity can be defined like foliated homogeneity using C∞foliated diffeomorphisms. Theorem B follows with an adaptation of an argument of Clark and Hurder [15, Theorem 5.2], using that the canonical left action of Homeo(X, F) on Xis micro-transitive by a theorem of Effros [20, 43]. To prove Theorem C, it is enough to show “(iii) ⇒(i)” by Theorem B. Assuming (iii), we get the so-called structural right local transverse action, which has its own interest; for instance, it was introduced and used in [7] for Lie foliations. It is the unique “foliated right local action up to leafwise homotopies” of Gon X, which corresponds to the local right translations on Gvia foliated charts (Proposition 6.6 and Section 6.3). Its construction uses a partition of unity subordinated to a foliated atlas and the leafwise center of mass for some (leafwise) Riemannian metric to merge the obvious right local transverse actions on the domains of foliated charts. The structural right local transverse action gives (i) because we always have leafwise homogeneity (Proposition 7.1). In Theorem C, our proof of “(iii) ⇒(i)” needs the C∞structure of X because we use the leafwise center of mass as an auxiliary tool. Of course, it could be possible to avoid the C∞condition and show “(iii) ⇒(ii)” directly with other tools, but that procedure would certainly require more work. Since there exist leaves without holonomy, and since the (differentiable) quasi-isometry type of the leaves is independent of the choice of a (leafwise) Riemannian metric on X, it follows that Xis not foliated homogeneous if there is a leaf with holonomy, or if there is a pair of non-quasi-isometric leaves. The reciprocal statement is not true in general. Fokkink and Oversteegen [23, Theorem 35] constructed an example of a non-homogeneous weak solenoid all of whose leaves are simply connected, and therefore it has no holonomy, and its leaves are quasi-isometric to each other because weak solenoids are suspension foliated spaces. Dyer, Hurder and Lukina constructed more examples of such weak solenoids [19, Theorem 10.7]. In Section 8.3, we give an example of a compact foliated space Xsatisfying the conditions of Theorem A, which is not foliated homogeneous and has no holonomy, whose leaves are quasi-isometric to each other, and with locally connected local transversals (thus it is not a weak solenoid). MOLINO’S DESCRIPTION AND FOLIATED HOMOGENEITY 5 2. Preliminaries See [39, Chapter II], [24] and [13, Chapter 11] for the needed preliminaries on foliated spaces and interesting examples, and [27, 28, 29] for the preliminaries on pseudogroups. We mainly follow [10, Sections 2 and 4A], which in turn follows [4, 5, 6]. Some ideas are also taken from [15, 9, 8]. The needed basic concepts and tools are recalled here for the reader’s convenience, and a few new observations are also made. In the whole paper, unless otherwise stated, spaces are assumed to be locally compact and Polish, and maps are assumed to be continuous. In particular, this applies to foliated spaces, topological groups, local groups and partial maps. 2.1. Pseudogroups. For spaces Tand T′, the notation φ∶T↣T′is used for a partial map. We will only consider the case where its domain, dom φ, is open in T. The germ of φat any u∈dom φwill be denoted by γ(φ, u). If φis an open embedding, we may identify φwith the homeomorphism φ∶domφ→imφof an open subset of Tto an open subset of T′, whose inverse can be considered as a partial map with open domain, φ−1∶T′↣T; in particular, when T=T′, such a φis called a local transformation of T. Given another space T′′, let Φ and Ψ be families of partial maps T↣T′ and T′↣T′′, respectively, with open domains. We use the notation ΨΦ = {ψφ ∣φ∈Ψ, ψ ∈Ψ}; in particular, Φn=Φ⋯Φ (ntimes) if T=T′and n∈Z+. If Φ consists of open embeddings, let Φ−1={φ−1∣φ∈Φ}. Recall that a pseudogroup Hon Tis a family of local transformations of Tthat contains idT, and is closed by the operations of composite, inversion, restriction to open sets and union. It is said that His generated by S⊂H if Hcan be obtained from Susing the above operations. By considering a pseudogroup as a direct generalization of a group of transformations, the basic dynamical concepts have obvious generalizations to pseudogroups, like orbits,saturation, (topological)transitivity and minimality. The orbit space is denoted by T/H. The H-saturation of any A⊂Tis denoted by H(A), and the orbit of any u∈Tby H(u). For any open V⊂T, the restriction H∣V∶={h∈H∣domh, imh⊂V}is a pseudogroup. Given another pseudogroup H′on T′, a morphism Φ∶H→H′is a maximal collection of partial maps T↣T′with open domain such that H′ΦH⊂Φ, T=⋃φ∈Φdomφ, and, for all φ, ψ ∈Φ and u∈dom φ∩dom ψ, there is some h′∈H′so that φ(u)∈domh′and γ(h′φ, u)=γ(ψ, u). Let Φ0be a family of partial maps T↣T′with open domain such that T=H(⋃φ∈Φdom φ), and there is a subset Sof generators of Hsuch that, if φ, ψ ∈Φ0,h∈Sand u∈domφ∩domψh, then there is some h′∈H′so that φ(u)∈dom h′and γ(h′φ, u)=γ(ψh, u). Then there is a unique morphism Φ ∶H→H′containing Φ0, which is said to be generated by Φ0. For instance, idTgenerates a morphism idH∶H→Hconsisting of all possible unions of maps in H; in particular, H⊂idH. For another pseudogroup H′′ on T′′ and a morphism Ψ∶H′→H′′, the family ΨΦ generates a morphism H→H′′, which may 6 J.A. ´ ALVAREZ L ´ OPEZ AND R. BARRAL LIJ ´ O be also denoted by ΨΦ with some abuse of notation. In this way, the morphisms of pseudogroups form a category PsGr. There is a canonical functor Top →PsGr, assigning the pseudogroup generated by idT, also denoted by T, to every topological space T, and assigning the morphism generated by φ, also denoted by φ, to every map φ∶T→T′. A morphism Φ ∶H→H′ is an isomorphism of PsGr if and only if it is generated by a family Φ0of open embeddings such that Φ−1 0generates a morphism H′→H, which is the inverse Φ−1in PsGr. With the terminology of Haefliger [27, 28, 29], an ´etal´e morphism Φ∶ H→H′is a maximal family of homeomorphisms of open subsets of Tto open subsets of T′such that H′ΦH⊂Φ, T=⋃φ∈Φdom φand ΦΦ−1⊂H′. If moreover Φ−1is an ´etal´e morphism, then Φ is called an equivalence, and the pseudogroups Hand H′are said to be equivalent. If Φ0is a family of homeomorphisms of open subsets of Tto open subsets of T′such that T= H(⋃φ∈Φdomφ)and Φ0HΦ−1 0⊂H′, then there is a unique ´etal´e morphism Φ ∶ H→H′containing Φ0, which is said to be generated by Φ0. Any equivalence generates an isomorphism in PsGr, and, vice versa, any isomorphism in PsGr is generated by a unique equivalence. Hence isomorphism and equivalences are equivalent concepts. Equivalent pseudogroups are considered to have the same dynamics. For instance, His equivalent to H∣Vfor any open V⊂T that meets all H-orbits. In fact, Φ ∶H→H′is an equivalence if and only if G=H∪H′∪Φ∪Φ−1is a pseudogroup on T⊔T′such that Tand T′meet all G-orbits, G∣T=Hand G∣T′=H′. The germs γ(h, u), for h∈Hand u∈dom h, form a topological groupoid H, equipped with the sheaf topology and the operation induced by composite. Its unit subspace can be identified with T. In fact, His an ´etal´e groupoid (the source and target maps, s, t ∶H→T, are local homeomorphisms). Given x∈T, the group of elements of γ∈Hwith s(γ)=t(γ)=xis called the germ group of Hat x. Let us recall the following definitions of properties that Hmay have: Compact generation: This means that there is a relatively compact open U⊂T, which meets all orbits, such that H∣Uis generated by a finite set, E={h1,...,hk}, and every hihas an extension ˜ hi∈Hwith domhi⊂dom ˜ hi. This Eis called a system of compact generation of Hon U. (Strong) equicontinuity: This means that there are an open cover {Ti}of Tand a metric diinducing the topology of every Ti, and His generated by some subset S⊂H, with S2⊂S=S−1(Sis symmetric and closed by composites1), such that, for every >0, there is some δ>0 so that di(x, y)<δÔ⇒dj(h(x), h(y))< 1The term pseudo∗group was used in [10] when these conditions are satisfied. This term was introduced in [36] for a family that moreover contains idTand is also closed by restrictions to open subsets. MOLINO’S DESCRIPTION AND FOLIATED HOMOGENEITY 7 for all h∈S, indices i, j, and x, y ∈Ti∩h−1(Tj∩im h). Strong quasi-analyticity: This means that His generated by some subset S⊂H, with S2⊂S=S−1, such that, if any h∈Sis the identity on some non-empty open subset of its domain, then h= iddom h. Strong local freeness: This means that His generated by some subset S⊂H, with S2⊂S=S−1, such that, if any h∈Sfixes some point in its domain, then h=iddom h. Equivalently, this means that His strongly quasi-analytic and all of its germ groups are trivial. These properties are invariant by equivalences. If compact generation holds with some U, then it also holds with any other relatively compact open subset of Tthat meets all orbits. Let Pdenote any of the above last three properties. If Pholds with S, then it also holds with its localization, Sloc ={h∣O∣h∈S, O is open in dom h}. Moreover we can add idTto Sif desired (obtaining S2=S). If His compactly generated and satisfies P, then, for every relatively compact open U⊂Tthat meets all orbits, we can choose a system of compact generation Eof Hon Usuch that H∣Ualso satisfies Pwith S=⋃∞ n=1En. The following result lists some needed non-elementary properties. Proposition 2.1 ([4, Proposition 8.9, and Theorems 11.1 and 12.1], [42] and [5, Theorems 3.3 and 5.2]).Suppose that His compactly generated, equicontinuous and strongly quasi-analytic. Then the following holds: (i) Assume that Hsatisfies the condition of compact generation with U, E={h1,...,hk}and ˜ h1,...,˜ hk. For every h=hin⋯hi1∈⋃∞ n=1En, let ˜ h=˜ hin⋯˜ hi1. Then there is a finite family Vof open subsets of Tcovering Usuch that, for any h∈⋃∞ n=1Enand V∈V, we have V⊂dom ˜ hif V∩domh≠∅. (ii) Suppose that Hsatisfies the equicontinuity condition with a set S. Then C(O, T )∩Sloc consists of local transformations for all small enough open subsets O⊂T, where the closure is taken in the compact-open topology, and the pseudogroup Hgenerated by such transformations is equicontinuous. More precisely, Hsatisfies the equicontinuity condition with the set Sdetermined by the condition C(O, T )∩S=C(O, T )∩Sloc for all Oas above. (iii) The orbit closures are minimal sets, and therefore His transitive if and only if it is minimal. In Proposition 2.1-(ii), the pseudogroup His called the closure of H. 2.2. Relation of pseudogroups with local groups and local actions. The general definition of local group is rather involved [34], but, in the locally compact case, a local group Gcan be considered as neighborhood of the identity element ein some topological group [16, 17]. Two such neighborhoods in the same topological group define equivalent local groups; 8 J.A. ´ ALVAREZ L ´ OPEZ AND R. BARRAL LIJ ´ O thus it can be said that, up to equivalences, a local group is the “germ” of a topological group at the identity element. For the sake of simplicity, the family of open neighborhoods of ein Gwill be denoted by N(G, e). Given another local group G′with identity element e′, a local homomorphism of G to G′is a partial map with open domain, σ∶G↣G′, such that e∈domσ, σ(e)=e′, and σ(gh)=σ(g)σ(h)for all g, h ∈domσsuch that the products gh and σ(g)σ(h)are defined with gh ∈domσ. Two local homomorphisms of Gto G′are equivalent when they have the same germ at e. If there is a local homomorphism τ∶G′↣Gsuch that τσ and στ are equivalent to idG and idG′, then σis called a local isomorphism. The term sublocal group will be used for a subspace H⊂Gsuch that (H∩V)2,(H∩V)−1⊂Hfor some V∈N(G, e); in particular, e∈H, but H∩Vis not required to be closed in V(contrary to [26, Definition 2.10.]). A sublocal group becomes a local group with the induced structure, but it may not be locally compact, and the inclusion map of any sublocal group is a local homomorphism. A right local action of Gon Tis a partial map with open domain, χ∶T×G↣T, where T×{e}⊂domχand χ(u, e)=ufor all u∈T, and such that, for all g, h ∈G and u∈T, if the product gh is defined and (u, g),(u, gh),(χ(u, g), h)∈ domχ, then χ(χ(u, g), h)=χ(u, gh). Two right local actions of Gon T are equivalent when they agree around T×{e}. If Tis compact, we can assume domχ=T×Ofor some O∈N(G, e). For any open V⊂T, the restriction χ∶χ−1(V)∩(V×G)→Vis a right local action of Gon V, called the restriction of χto V. Given an open cover {Ti}of Tand a right local action χiof Gon every Tisuch that the restrictions of χiand χjto Ti∩Tj are equivalent, it is easy to check that there is a unique right local action of Gon T, up to equivalences, whose restriction to every Tiis equivalent to χi. Consider another right local action χ′of G′on T′. A partial map with open domain, φ∶T↣T′, is called locally equivariant if there is some open neighborhood Σ of dom φ×{e}in dom χ∩(φ×idG)−1(dom χ′)such that χ(Σ)⊂domφand φχ(u, g)=χ′(φ(u), g)for all (u, g)∈Σ. Note that composites, restrictions to open sets and unions of locally equivariant partial maps with open domain are locally equivariant, as well as their inverses whenever defined. A family of partial maps T↣T′with open domain is called locally equivariant when all of its elements are locally equivariant. Local anti-homomorphisms,left local actions, their equivalences and corresponding locally equivariant maps are similarly defined. For instance, any finite dimensional metrizable locally compact local group is indeed locally isomorphic to the direct product of a Lie group and a compact zero-dimensional topological group [34, Theorem 107] (corrected according to [26], or using [16, 17] and [38, Section IV.4.9]). As a concrete example, we can consider the product of any local Lie group and any countable family of finite groups. By Ado’s theorem, the equivalence classes of local Lie groups and their local homomorphisms correspond one-to-one to finite dimensional real Lie algebras and their homomorphisms. A typical example of right local action of a local group Gon itself is given by its MOLINO’S DESCRIPTION AND FOLIATED HOMOGENEITY 9 local right translations, and any local left translation of Gbecomes locally equivariant. Proposition 2.2 ([5, Theorems 3.3 and 5.2], [10, Lemma 2.36, Theorem 2.38 and Remark 21]).The following holds: (i) Suppose that His minimal, compactly generated, equicontinuous and strongly quasi-analytic. Then His strongly locally free if and only if His equivalent to a pseudogroup on some local group G generated by the left local action by local left translations of a finitely generated dense sublocal group Γ⊂G. (ii) Let Gand G′be the pseudogroups on local groups Gand G′generated by the left local actions by local left translations of respective finitely generated dense sublocal groups Γand Γ′. Let Φ∶G→G′be a morphism such that G(e)↦G′(e′)by the induced map G/G→G/G′. Then Φis generated by a local homomorphism G↣G′that restricts to a local homomorphism Γ↣Γ′. Proposition 2.3. Let Φ∶H→H′be an equivalence between compactly generated pseudogroups. Let χbe a right local action of Gon Tsuch that His locally equivariant. Then there is a unique right local action χ′of G on T′, up to equivalences, such that Φand H′are locally equivariant. Proof. Let Ebe a system of compact generation of Hon a relatively compact open U⊂T, and let ˜ hbe an extension of every h∈Ewith dom h⊂dom ˜ h. There is a subset Φ0⊂Φ such that {dom φ×im φ∣φ∈Φ0}covers U×T′, {imφ∣φ∈Φ0}is locally finite in T′, and every φ∈Φ0has an extension ˜ φ∈Φ with domφ⊂dom ˜ φ. Write {φi}={φh ∣h∈E, φ ∈Φ0}, and let ˜ φi=˜ φ˜ hif φi=φh for h∈Eand φ∈Φ0. Moreover let Ui=dom φi, U′ i=imφi, Ui=dom ˜ φi, U′ i=im ˜ φi, Uij =˜ φ−1 j( U′ i∩ U′ j)=dom ˜ φ−1 i˜ φjand  U′ ij =φj( Ui∩ Uj)=dom ˜ φi˜ φ−1 j. The following assertion is easy to check. Claim 1.{φi}generates Φ and {φiφ−1 j}generates H′. Let Ω =domχ, and let Σij be an open neighborhood of  Uij ×{e}in Ω ∩ (˜ φ−1 i˜ φj×idG)−1(Ω)such that χ(Σij)⊂ Uij and ˜ φ−1 i˜ φjχ(u, g)=χ(˜ φ−1 i˜ φj(u), g) for all (u, g)∈Σij. Let Ω′ 0={(u′, g)∈T′×G∣u′∈U′ i∩U′ j⇒(˜ φ−1 j(u′), g)∈Σij,∀i, j }. Claim 2.Ω′ 0is open in T′×G. Take some (u′, g)∈Ω′ 0. Let Ibe the set of indices isuch that u′∈U′ i, and let I′be the set of pairs of indices, (i, j), such that u′∈U′ i∩U′ j, which are finite sets because {U′ i}is locally finite in T′. Then, using that U′ i⊂ U′ i, every ˜ φiis a homeomorphism, and Σij is an open neighborhood of (˜ φ−1 j(u′), g)in  Uij ×Gfor all (i, j)∈I′, it follows that there are open neighborhoods, Vof u′in T′and Pof gin G, such that V∩U′ i=∅if i/∈I, and ˜ φ−1 j(V)×P⊂Σij for all (i, j)∈I′. Thus V×P⊂Ω′ 0. 16 J.A. ´ ALVAREZ L ´ OPEZ AND R. BARRAL LIJ ´ O defined by adding the condition p′ aiφ=p′ aiψon every Kito the above definition of Nk F(φ, U,U′,K,E); using (3), this extra condition can be also written as φ2 aii=ψ2 aiion pi(Ki)for all i. The weak plaquewise Crtopology is similarly defined by requiring the conditions only for finite families of indices i. The subindex “WP/SP” will be added to the notation to indicate that the weak/strong plaquewise Crtopology is considered in a family of Crfoliated maps. Note that, if two foliated maps are close enough in Cr SP(X, F;X′,F′), then they induce the same morphism H→H′; in fact, they are leafwisely homotopic if r=∞, as follows by taking basic open sets Nk P(φ, U,U′,K,E) as above where the plaques of the foliated charts in U′are convex balls in the leaves for a given Riemannian metric on X′, and then using geodesic segments to define homotopies. With the strong plaquewise Crtopology, we can continue the direct extensions of results about spaces of Crmaps between manifolds. Proposition 2.6. The following properties hold: (i) Embr(X, F;X′,F′)is open in Cr SP(X, F;X′,F′)for 1≤r≤∞. (ii) For 1≤r≤∞, the set of closed Crfoliated embeddings is open in Cr SP(X, F;X′,F′). (iii) Diffeor(X, F;X′,F′)is open in Cr SP(X, F;X′,F′)for 1≤r≤∞. (iv) Cs(X, F;X′,F′)is dense in Cr SP(X, F;X′,F′)for 0≤r<s≤∞. (v) Diffeos(X, F;X′,F′)is dense in Diffeor SP(X, F;X′,F′)for 1≤r< s≤∞. (vi) If 1≤r<∞, any Crfoliated space is Crdiffeomorphic to a C∞ foliated space. (vii) If 1≤r<s≤∞, two Csfoliated spaces are Csdiffeomorphic if and only if they are Crdiffeomorphic. Proof. Adapt the proofs of [32, Theorems 2.1.4, 2.1.6, 2.2.6, 2.2.7, 2.2.9 and 2.2.10, and Corollary 2.1.6].  Like in the case of manifolds, it easily follows from Proposition 2.6-(iv) that, for 0 ≤r<s≤∞, if there is a Crleafwise homotopy between Cs foliated maps, then there is a Csleafwise homotopy between them. The above openness statements are stronger with the strong foliated Crtopology, whereas the denseness statements are stronger for the strong plaquewise Crtopology. There is no version of Proposition 2.6-(i) with the strong foliated Crtopology (for instance, consider the case of compact spaces foliated by points). However we can prove a weaker form of that statement by using certain subspaces Cr SF(X, F;X′,F′)defined as follows. A foliated map φ∶X→X′is called a transverse embedding (respectively, transverse equivalence) if the induced morphism Φ ∶H→H′is generated by embeddings (respectively, Φ is an isomorphism). Observe that F′(φ(X))=X′if φis a transverse equivalence. A subset M⊂C(X, F;X′,F′)of transverse embeddings (respectively, transverse equivalences) is called uniform if there are some foliated atlases, Uof Xand U′of X′like in Section 2.3, such MOLINO’S DESCRIPTION AND FOLIATED HOMOGENEITY 17 that, for all φ∈M, the maps φ2 ai in (3) are embeddings (respectively, open embeddings). Note that, if these properties hold with Uand U′, then they hold with all finer atlases. For example, Emb(X, F;X′,F′)consists of uniform transverse embeddings, and Homeo(X, F;X′,F′)consists of uniform transverse equivalences. Proposition 2.7. For 1≤r≤∞, let M⊂Cr SF(X, F;X′,F′)be a uniform subspace of transverse embeddings. Then Embr(X, F;X′,F′)∩Mis open in M. Proof. It is enough to prove the case r=1. For any φ∈Emb1(X, F;X′,F′)∩ M, consider a basic open set N1∶=N1 F(φ, U,U′,K,E)in C1 SF(X, F;X′,F′) as above. We can assume that K(and therefore U) covers X, and U′covers X′. After refinements, we can choose U,U′and Ksuch that the maps ψ2 ai are embeddings for all ψ∈M, and the interiors Vi∶=˚ Kicover X. Take an open cover {Wi}of Xwith Wi⊂Vifor all i. By [32, Lemma 1.3], we can choose Esuch that the maps ψ∶p−1 i(u)∩Vi→p′−1 ai(ψ2 aii(u)) are C1 embeddings for u∈pi(Vi)and ψ∈N1. Hence ψ∶Vi→X′is a C1foliated embedding for all ψ∈N1∩M. Now, we adapt the final part of the proof of [32, Theorem 1.4] as follows. Since φis an embedding, we get disjoint open subsets V′ i, W ′ i⊂X′for every isuch that φ(Wi)⊂W′ iand φ(X∖Vi)⊂V′ i. Then it is easy to find a neighborhood N0of φin CSF(X, F;X′,F′)so that ψ(Wi)⊂W′ iand ψ(X∖ Vi))⊂V′ ifor all ψ∈N0. We finally obtain N0∩N1∩M⊂Emb1(X, F;X′,F′).  Proposition 2.8. For 1≤r≤∞, let M⊂Cr SF(X, F;X′,F′)be a uniform subspace of transverse equivalences. Then Diffeor(X, F;X′,F′)∩Mis open in M. Proof. We adapt the proofs of [32, Corollary 1.6 and Theorem 1.6]. The set M′={φ∈Propr(X, F;X′,F′)∣Txφis surjective ∀x∈X} is closed in Cr SF(X, F;X′,F′)by Proposition 2.5-(i),(ii). On the other hand, Embr(X, F;X′,F′)∩Mis open in Mby Proposition 2.7. Thus the result follows because Embr(X, F;X′,F′)∩M′=Diffeor(X, F;X′,F′). According to Proposition 2.6-(vi),(vii), we will only consider either (C0) foliated spaces or C∞foliated spaces from now on. Proposition 2.9. Let φ∶X→X′be a foliated map. Suppose that X′is equipped with a C∞structure. Then there is at most one C∞structure on Xsuch that φis C∞and Txφis an isomorphism for all x∈X. Proof. Consider two C∞structures on X, and take C∞foliated charts, ξ1∶ U1→B1×T1of the first C∞structure on X,ξ2∶U2→B2×T2of the second C∞structure on X, and ξ′∶U′→B′×T′of the C∞structure on X′. We 18 J.A. ´ ALVAREZ L ´ OPEZ AND R. BARRAL LIJ ´ O can assume that U2⊂U1and φ(U1)⊂U′. Then ξ′φξ−1 1(v1, u1)=(g′ 1(v1, u1), h′ 1(u1)), ξ′φξ−1 2(v2, u2)=(g′ 2(v2, u2), h′ 2(u2)), ξ1ξ−1 2(v2, u2)=(g12(v2, u2), h12(u2)), for (vk, uk)∈Bk×Tk,k=1,2, where g′ k∶Bk×Tk→B′has partial derivatives of arbitrary order with respect to vk, continuous on Bk×Tk, and g12 ∶B2× T2→B1is continuous. Moreover the differential map of g′ 1with respect to v1 is an isomorphism at any point. Therefore, by the inverse function theorem, we can assume that g′ 1(⋅, u1)∶B1→g′ 1(B1×{u1})is a C∞diffeomorphism for all u1∈T1. Its inverse function is denoted by ¯g′ 1(⋅, u1)∶g′ 1(B1×{u1})→B1. For any small ball B′ 0⊂B′, let T10 ⊂T1be the open subset that consists of the points u1∈T1such that B′ 0⊂g′ 1(B1×{u1}). It also follows from the inverse function theorem that the partial derivatives of arbitrary order of ¯g′ 1(⋅, u1)∶B′ 0→B1depend continuously on u1. Since g12(v2, u2)=¯g1(g2(v2, u2), h12(u2)) on B2×h21(T10), the function g12 ∶B2×h21(T10)→B1has partial derivatives of arbitrary order with respect to v2, continuous on B2×h21(T10). 2.5. Center of mass. In Section 6.2, we will use the center of mass of a mass distribution on a Riemannian manifold M[35], [14, Section IX.7]. Let Ω ⊂Mbe a compact submanifold with boundary with dim Ω =dim M. For 0 ≤r≤∞, let C(Ω)be the set of functions f∈Cr+2(Ω)such that grad fis an outward pointing vector field on ∂Ω and Hess fis positive definite on the interior ˚ Ω of Ω. Note that C(Ω)is open in the Banach space Cr+2(Ω)with the norm ∥∥Cr+2,Ω,g, and therefore it is a C∞Banach manifold. Moreover C(Ω)is preserved by the operations of sum and product by positive numbers. Any f∈C(Ω)attains its minimum value at a unique point mΩ(f)∈˚ Ω, defining a function mΩ∶C(Ω)→˚ Ω. Lemma 2.10 ([3, Lemma 10.1 and Remark 11-(ii)]).The map mΩis Cr. Suppose that Mis connected and complete. Let (A, µ)be a probability space, Ba convex open ball of radius r>0 in M, and f∶A→Ba measurable map, which is called a mass distribution on B. Consider the C∞function Pf,µ ∶B→Rdefined by Pf,µ(x)=1 2∫Ad(x, f(a))2µ(a). Proposition 2.11 (H. Karcher [35, Theorem 1.2]).We have the following: (i) gradPf,µ is an outward pointing vector field on the boundary ∂B. (ii) If δ>0is an upper bound for the sectional curvatures of Min B, and 2r<π/2√δ, then HessPf,µ is positive definite on B. MOLINO’S DESCRIPTION AND FOLIATED HOMOGENEITY 19 If the hypotheses of Proposition 2.11 are satisfied, then Pf,µ ∈C(B), and therefore Cf,µ ∶=mB(Pf,µ)∈Bis defined and called the center of mass of f (with respect to µ). This point is independent of the choice of Bsatisfying the above conditions. The following is a consequence of Lemma 2.10. Corollary 2.12 ([3, Corollary 10.3]; cf. [35, Corollary 1.6]).The following properties hold: (i) Cf,µ depends continuously on fand the metric tensor of M. (ii) If Ais the Borel σ-algebra of a metric space, then Cf,µ depends continuously on µin the weak-∗topology. Consider the following particular case. Let Nbe a C∞manifold, φ= (φ1,...,φk)∶N→MkaC∞map, and λ=(λ1,...,λk)a finite C∞partition of unity of N. For every x∈N, consider the probability measure µφ,λ,x = ∑k i=1λi(x)δφi(x), where δydenotes the Dirac mass at every y∈M. Suppose that, for all x∈N, the points φ1(x),...,φk(x)lie in a ball Bxof Msatisfying the conditions of Proposition 2.11. Then we can define center of mass Cφ,λ,x of idBxwith respect to µφ,λ,x, which is independent of the choice of Bx. The following sharpening of Corollary 2.12 also follows from Lemma 2.10. Corollary 2.13. The map N→M,x↦Cφ,λ,x, is C∞. 3. Molino’s description Consider the notation of Section 2.3 in the rest of the paper. Proof of Theorem A. Most of the properties stated in this theorem were already proved in [10, Theorem A]. It only remains to prove the part concerning H. For this purpose, we have to recall the construction of G, X0,  F0and ˆπ0. We can assume that Xsatisfies the conditions of equicontinuity and strong quasi-analyticity with the same set S, and that Hsatisfies the conditions of equicontinuity and strong quasi-analyticity with the induced set S. Let Sc-o be the space Swith the restriction of the compact-open topology on the set of partial maps T↣Twith open domain [1]. Consider the subspace Sc-o ∗T={(g, u)∈S×T∣u∈domg}⊂Sc-o ×T , and equip the set  Tof all germs of maps in S(or H) with the final topology induced by the germ map γ∶Sc-o ∗T→ T(this is not the restriction of the sheaf topology). Consider the restrictions s, t ∶ T→Tof the sourse and target maps. The space  Tis locally compact and Polish, and ˆπ∶=(s, t)∶  T→T×Tis continuous and proper. Fix some point u0∈Ti0⊂T. Then the subspace  T0∶=s−1(u0)⊂ Tis locally compact and Polish. This definition is different from the one given in [10, Section 3D], where  T0=t−1(u0)was considered. This change can be made because the inversion of local transformations defines a homeomorphism of Sc-o [10, Proposition 3.1], and therefore the germ inversion defines 20 J.A. ´ ALVAREZ L ´ OPEZ AND R. BARRAL LIJ ´ O a homeomorphism of  T, which becomes a topological groupoid by [1, Proposition 10]. The rest of definitions and arguments of [10, Sections 3D–3G] must be changed accordingly. For instance, take ˆπ0=t∶ T0→T(instead of ˆπ0=s, used in [10]), which is open, continuous and proper, and its fibers are homeomorphic to each other [10, Section 3D]. We have  T0≡⊔i Ti,0, where  Ti,0=ˆπ−1 0(Ti). Note that H∶=ˆπ−1 0(u0)=ˆπ−1(u0, u0)becomes a compact Polish group since  Tis a topological groupoid. Moreover the germ product defines a continuous free right action of Hon  T0whose orbits are clearly equal to the fibers of ˆπ0∶ T0→T. Thus this map induces a continuous bijection  T0/H→T. In fact this bijection is a homeomorphism, as easily follows by using also that His compact,  T0is locally compact, and Tis Hausdorff. For any h∈H, define ˆ h∶ˆπ−1 0(domh)→ˆπ−1 0(imh)by ˆ h(γ(g, u0)) = γ(hg, u0)for g∈Swith u0∈domgand g(u0)∈domh(instead of ˆ h(γ(g, u))= γ(gh−1, h(u))for u∈dom g∩dom hwith g(u)=u0, used in [10]). The maps ˆ hare local transformations of  T0satisfying hˆπ0=ˆπ0ˆ h, idT=id T0, hh′=ˆ h h′ and ˆ h−1= h−1[10, Sections 3E]. Moreover it is easy to see that every ˆ his H-equivariant (note that domˆ hand im ˆ hare H-invariant). Let  H0be the pseudogroup on  T0generated by  S0={ˆ h∣h∈S}. There is a local group Gand some dense finitely generated sublocal group Γ ⊂Gsuch that  H0is equivalent to the pseudogroup generated by the local action of Γ on Gby local left translations [10, Proposition 3.41]—this was proved by checking that  H0is compactly generated, equicontinuous and strongly locally free, and its closure is also strongly locally free, and then applying Proposition 2.2-(i). Furthermore ˆπ0generates a morphism  H0→H. Let ˇ Ui,0=Ui× Ti,0×{i}≡Ui× Ti,0, equipped with the product topology, and consider the topological sum ˇ X0∶=⊔ i(Ui× Ti,0)=⋃ i ˇ Ui,0, and the closed subspaces  Ui,0∶={(x, γ, i)∈ˇ Ui,0∣pi(x)=ˆπ0(γ)}⊂ˇ Ui,0, X0∶=⋃ i Ui,0⊂ˇ X0. Note that  X0is the topological sum of the spaces  Ui,0. Consider the equivalence relation “∼” on  X0defined by (x, γ, i)∼(y, δ, j)if x=yand γ= hji(δ). Let  X0be the corresponding quotient space, let q∶ X0→ X0be the quotient map, let [x, γ, i]=q(x, γ, i), let  Ui,0=q( Ui,0), and let ˜pi,0∶ Ui,0→ Ti,0 denote the restriction of ˇpi,0∶ˇ Ui,0≡Ui× Ti,0→ Ti,0, which induces a map ˆpi,0∶ Ui,0→ Ti,0. Moreover a map ˆπ0∶ X0→Xis defined by ˆπ0([x, γ, i])=x. Observe that  Ui,0=ˆπ−1 0(Ui). Then  X0is compact and Polish, { Ui,0,ˆpi,0, hij} is a defining cocycle of a minimal foliated structure  F0on  X0, ˆπ0is continuous and open, the fibers of ˆπ0are homeomorphic to each other, and the restriction of ˆπ0to the leaves of  X0are the holonomy coverings of the leaves MOLINO’S DESCRIPTION AND FOLIATED HOMOGENEITY 21 of X[10, Section 4B]. In the proof of these properties, it was used that every restriction q∶ Ui,0→ Ui,0is a homeomorphism. Since every  Ti,0is H-invariant, we get an induced free right action of H on every ˇ Ui,0≡Ui× Ti,0, acting as the identity on the factor Ui, yielding a right H-action on ˇ X0by union. This restricts to a free right action of H on  X0, preserving every  Ui,0, because the H-orbits in  T0are equal to the fibers ˆπ0∶ T0→T. Since moreover every  hij is H-equivariant, we get an induced right action on  X0, given by [x, γ, i]⋅σ=[x, γσ, i]for [x, γ, i]∈ X0 and σ∈H. This action is also free because every restriction q∶ Ui,0→ Ui,0 is a homeomorphism, and it is easy to see that its orbits equal the fibers of ˆπ0∶ X0→X. Finally note that every map ˆpi,0∶ Ui,0→ Ti,0is H-equivariant, and therefore Hacts on  X0by foliated transformations.  In the rest of this section, assume that Xsatisfies the hypotheses of Theorem A. Consider structures (G, H,  X0,ˆπ0)satisfying the conditions of its statement, where  X0is considered as a foliated space and H-space. If desired, we may also add a finitely generated dense sublocal group Γ ⊂ Gto the notation, (G, Γ, H,  X0,ˆπ0), so that the holonomy pseudogroup of  X0is represented by the pseudogroup generated by the left local action of Γ on Gby local left translations. It is said that two such structures, (G, Γ, H,  X0,ˆπ0)and (G′,Γ′, H′, X′ 0,ˆπ′ 0), are equivalent if there are a local isomorphism ψ∶G↣G′that restricts to a local isomorphism Γ ↣Γ′, an isomorphism χ∶H→H′, and a foliated χ-equivariant homeomorphism φ∶ X0→ X′ 0such that ˆπ0=ˆπ′ 0φ(the condition on Γ and Γ′is omitted if Γ and Γ′are not considered). In this case, (ψ, χ, φ)is called an equivalence. This notion of equivalence is natural because it clearly means that the descriptions of the foliated space Xgiven by (G, Γ, H,  X0,ˆπ0)and (G′,Γ′, H′, X′ 0,ˆπ′ 0)are essentially the same, giving rise to equivalent invariants of X. For instance, G, Γ and Hhave the same algebraic and topological properties as G′, Γ′ and H′, and ˆπ0is a principal bundle projection if and only is ˆπ′ 0is also a principal bundle projection. Proposition 3.1 (Cf. [10, Propositions 3.43, 4.12 and 4.13]).All structures (G, Γ, H,  X0,ˆπ0)constructed in the proof of Theorem A are equivalent. Proof. We have to prove that the equivalence class of (G, Γ, H,  X0,ˆπ0)is independent of the choices of u0,Sand {Ui, pi, hij}. Most of this is already proved in [10, Propositions 3.43, 4.12 and 4.13]. We only have to check what concerns H. To begin with, take another point of u1∈Ti1⊂T, and let  T1, ˆπ1, S1,  H1,G1, Γ1and H1be constructed like  T0, ˆπ0, S0, H0,G0∶=G, Γ0∶=Γ and H0∶=Hby using u1instead of u0. Now, for each h∈H, let us use the notation ˆ h0∶=ˆ h∈ H0, and let ˆ h1∶ˆπ−1 1(domh)→ˆπ−1 1(imh)be the map in  H1defined like ˆ h. In particular, the maps ( hij)1are defined like the maps ( hij)0∶= hij. There is some f0∈Ssuch that u0∈domf0and 22 J.A. ´ ALVAREZ L ´ OPEZ AND R. BARRAL LIJ ´ O f0(u0)=u1. Let θ∶ T0→ T1be defined by θ(γ(f, u0))=γ(ff−1 0, u1)(instead of θ(γ(f, x))=γ(f0f, x), like in [10]). This map is a homeomorphism, and satisfies ˆπ0=ˆπ1θ, dom ˆ h1=θ(dom ˆ h0)and ˆ h1θ=θˆ h0for all h∈S, obtaining that θgenerates an equivalence Θ ∶ H0→ H1[10, Proposition 3.42]. For k= 0,1, let Gkbe the pseudogroup on Gkgenerated by local left translations by elements of Γk. Via equivalences  Hk→Gk, Θ corresponds to an equivalence Θ′∶G0→G1. Since the local right translations of G1generate equivalences of G1, we can assume that the orbits of the identity elements correspond by the induced map G0/G0→G1/G1. By Proposition 2.2-(ii), it follows that Θ′is generated by a local isomorphism ψ∶G0↣G1that restricts to a local isomorphism Γ ↣Γ′. On the other hand, the conjugation mapping, γ(f, u0)↦γ(f0ff−1 0, u1), defines an isomorphism χ∶H0→H1so that θis χ-equivariant. Now, define  X1≡( X1, F1),[x, γ, i]1and ˆπ1∶ X1→Xlike  X0≡( X0, F0), [x, γ, i]0∶=[x, γ, i]and ˆπ0∶ X0→X, using  T1, ˆπ1∶ T1→Tand the maps ( hij)1instead of  T0, ˆπ0∶ T0→Tand the maps ( hij)0. According to [10, Proposition 4.12], a foliated homeomorphism φ∶ X0→ X1is defined by φ([x, γ, i]0)=[x, θ(γ), i]1, which satisfies ˆπ0=ˆπ1φand induces the equivalence Θ ∶ H0→ H1. Moreover φis χ-equivariant: for all [x, γ, i]0∈ X0and σ∈H0, φ([x, γ, i]0⋅σ)=φ([x, γσ, i]0)=[x, θ(γσ), i]1 =[x, θ(γ)χ(σ), i]1=[x, θ(γ), i]1⋅χ(σ). All choices of Sdefine the same space  T0by [10, Propositions 3.43], giving rise to the same Molino’s description. To prove the independence of {Ui, pi, hij}, it is enough to consider the case where {Ui, pi, hij}refines another defining cocycle {U′ a, p′ a, h′ ab}. Let H′be the corresponding representative of the holonomy pseudogroup on T′=⊔aT′ a. If Ui⊂U′ ai, there is an induced open embedding φi∶Ti→T′ ai. These maps generate an equivalence Φ ∶H→H′. In fact, h′ aiajφj=φihij. Let u′ 0=φi0(u0)∈T′ ai0⊂T′, and let S′⊂H′be a generating subset such that S′2⊂S′=S′−1. We can also use {U′ a, p′ a, h′ ab},u′ 0and S′to define  T′ 0, ˆπ′ 0∶ T′ 0→T′and  H′ 0like  T0, ˆπ0∶ T0→Tand  H0; in particular, the generators  h′ ab of  H′ 0are defined like the generators  hij of  H0. We get open embeddings ˆ φi,0∶ Ti,0→ T′ ai,0defined by ˆ φi,0(γ(g, u0))=γ(φigφ−1 i0, u′ 0), which generate an equivalence  Φ0∶ H0→ H′ 0(this is a corrected version of [10, Proposition 3.44]). Let (G′,Γ′, H′, X′ 0,ˆπ′ 0)be the Molino’s description defined with  T′ 0, ˆπ′ 0∶ T′ 0→T′and the maps  h′ ab. Let us use the notation [x, γ′, a]′for the element of  X′ 0represented by a tern (x, γ′, a). Let Gand G′be the pseudogroups on Gand G′generated by the local left translations by elements of Γ and Γ′. Via equivalences H→Gand H′→G′, Φ0corresponds to an equivalence  Φ′ 0∶G→G′. As above, we can assume that the orbits of the identity elements correspond by the induced map G/G→G′/G′, MOLINO’S DESCRIPTION AND FOLIATED HOMOGENEITY 23 and therefore, according to Proposition 2.2-(ii),  Φ′ 0is generated by a local isomorphism ψ∶G↣G′that restricts to a local isomorphism Γ ↣Γ′. Moreover ˆ φi0,0restricts to an isomorphism χ∶H→H′so that any map in  Φ0is χ-equivariant. Finally, a canonical foliated homeomorphism φ∶ X0→ X′ 0is well defined by φ([x, γ, i])=[x, ˆ φi,0(γ), ai]′[10, Proposition 4.13]. It is easy to check that φis H-equivariant.  By Proposition 3.1, the equivalence class of any structure (G, Γ, H,  X0,ˆπ0) constructed in the proof of Theorem A can be called the Molino’s description of X. According to the discussion of [10, Section 1.E], these structures are kind of a topological interpretation of the original Molino’s description in the case of a Riemannian foliation. That similarity can be indeed realized as an equivalence between the original Molino’s description and ours in that case. According to Molino’s terminology, the local isomorphism class of Gis called the structural local group [10], and, with the terminology of [18, 19],  X0will be called the Molino space and Hthe discriminant group. Proposition 3.2. Xis a G-foliated space for some local group Gif and only if its discriminant group is trivial. Proof. The “if” part of the statement is directly given by Theorem A. To prove the “only if” part, assume Xis a G-foliated space for some local group G. Thus His strongly locally free, obtaining that H={e}according to the definition of Hgiven in the proof of Theorem A.  For every ˆx∈ X0, let  Lˆxdenote the leaf of  X0through ˆx, and consider the identity  Lhol x≡ Lˆxgiven by Theorem A. Lemma 3.3. For x∈Xand ˆx≡[x, γ, i]∈ˆπ−1 0(x), let c∶I→Xbe a leafwise path from xto some point y, and let ˆcbe the unique lift of cto  Lhol x≡ Lˆxbeginning at ˆx. Then ˆc(1)≡[y, δγ, jβ], where δ=γ(hJ, pi(x))for any J=(j0,...,jβ)covering cwith j0=i. Proof. Take a partition 0 =t0<t1<⋅⋅⋅<tβ+1=1 of Isuch that c([tk, tk+1])⊂ Ujkfor k=0,...,β. For s∈I, the path cs(t)∶=c(st)in Lis covered by Js∶=(j0,...,jβs), where βs=min{k∈{0,...,β} ∣ tk+1≥s}, and let δs=γ(hJs, pi(x)). Then it is easy to see that ˆc(s)=[c(s), δsγ, jβs]. Fix some point x0∈p−1 i0(u0)⊂Ui0⊂X. Proposition 3.4. For ˆx0∈ˆπ−1 0(x0), we have Hol(Lx0, x0)={γ∈H∣ Lˆx0⋅γ= Lˆx0},(4) and the map  Lhol x0≡ Lˆx0↪ X0becomes equivariant with respect to the homomorphism Hol(Lx0, x0)↪H. 24 J.A. ´ ALVAREZ L ´ OPEZ AND R. BARRAL LIJ ´ O Proof. Consider the notation of the proof of Theorem A. Observe that Hol(Lx0, x0)is a subgroup of H: Hol(Lx0, x0)={γ(h, u0)∣h∈H, u0∈domh, h(u0)=u0} ⊂H={γ(g, u0)∣g∈H, u0∈domg, g(u0)=u0}. If γ=hol([c])∈Hol(Lx0, x0)for some [c]∈π1(Lx0, x0), then γ=γ(h−1 I, u0) for some I=(i0, i1,...,iα)covering cwith iα=i0. For any y∈Lx0and ˆy∈ Lˆx0∩ˆπ−1 0(y), we have ˆy≡[y, δ, i], where δ=γ(hJ, u0)for some admissible sequence J=(j0,...,jβ), with j0=i0and jβ=i, which covers a leafwise path cy∶I→Xfrom x0to y. Then ˆy⋅γis the final point of the lift to  Lhol x≡ Lˆx0, beginning at ˆy, of the loop c−1 yccyin Lx0, based at y. Thus ˆy⋅γ≡[y, δγδ−1δ, i0]=[y, δγ, i0]=[y, δ, i0]⋅γ , where the identity between these elements of  Lhol xand  Lˆx0is given by Lemma 3.3, applied to ˆyand c−1 yccy, because J−1IJ is defined and covers c−1 yccy, and hJ−1IJ =hJhIh−1 J. This proves the inclusion “⊂” in (4) and the equivariance of  Lhol x0≡ Lˆx0↪ X0. On the other hand, since the right H-action on  X0is free, foliated and preserves every ˆπ0-fiber, any element of the right hand side of (4) defines a covering transformation of the restriction ˆπ0∶ Lˆx0→Lx0, showing the inclusion “⊃” in (4).  According to the proof of Proposition 3.1, it follows from Proposition 3.4 that, for all x∈Xand ˆx∈ˆπ−1 0(x), there is an isomorphism Hol(Lx, x)≅{γ∈H∣ Lˆx⋅γ= Lˆx} so that the map  Lhol x≡ Lˆx↪ X0becomes equivariant with respect to the induced injective homomorphism Hol(Lx, x)→H. Nevertheless this isomorphism is not canonical in general. 4. Foliated homogeneous foliated spaces The foliated space Xis called foliated homogeneous when the canonical left action of Homeo(X, F)on Xis transitive. Similarly, if Xis C∞, it is called C∞foliated homogeneous when the canonical left action of Diffeo(X, F)on Xis transitive. A priory, C∞foliated homogeneity is stronger than foliated homogeneity, but we will see that indeed they are equivalent conditions for compact minimal C∞foliated spaces (Section 7). Take any complete metric dinducing the topology of X, and let Dbe the induced complete metric on Homeo(X)defined by D(φ, ψ)=sup x∈X d(φ(x), ψ(x))+sup x∈X d(φ−1(x), ψ−1(x)). In this way, Homeo(X)becomes a completely metrizable topological group, and its canonical left action on Xis continuous. Moreover it is easy to check that Homeo(X, F)is closed in Homeo(X), and therefore Homeo(X, F)is also a completely metrizable topological group. MOLINO’S DESCRIPTION AND FOLIATED HOMOGENEITY 25 Suppose that Xis compact. Then Dinduces the compact-open topology on Homeo(X), as follows from [11, Theorem 3], obtaining that Homeo(X) is also second countable. So Homeo(X)is a Polish group, and Homeo(X, F) a Polish subgroup. Therefore, by a theorem of Effros [20, 43], if Xis foliated homogeneous, then the canonical left action of Homeo(X, F)on X is micro-transitive; i.e., for all x∈Xand any neighbourhood Nof idXin Homeo(X, F), the set N⋅xis a neighborhood of xin X. Proof of Theorem B. Clark and Hurder have proved that any C∞homogeneous matchbox manifold is equicontinuous [15, Theorem 5.2]. Indeed, their argument applies to any compact minimal foliated homogeneous foliated space. Moreover the C∞structure is not used in that result. Thus the conditions of our statement are enough to get that (X, F)is equicontinuous. The rest of the proof uses the same main tool as in [15, Theorem 5.2], the indicated theorem of Effros. Let us prove that His strongly locally free. Since {Ui}is finite, there is some >0 such that d(Ui, X ∖ Ui)<for all i. Since the action of Homeo(X, F)on Xis micro-transitive, there is some δ>0 such that, for all x, y ∈Xwith d(x, y)<δ, there exists some φ∈Homeo(X, F)so that D(φ, idX)<and φ(x)=y. Since every Tihas compact closure in  Ti, we easily get a finite open cover {Tia}of Tisuch that the d-diameter of every σi(Tia)is smaller than δ. Let Uia =ξ−1 i(Bi×Tia),ξia =ξi∣Uia , Uia = Uiand ˜ ξia =˜ ξi. By using {Uia, ξia}and { Uia,˜ ξia}, varying iand a, instead of {Ui, ξi}and { Ui,˜ ξi}, it follows that we can assume that the d-diameter of every σi(Ti)is smaller than δ. Take Sequal to the family of the maps hIfor admissible sequences I. Suppose that some hI∈Sfixes a point u∈dom hI. Thus I=(i0,...,iα) with iα=i0. Let x=σi0(u)∈Ui0and let c∶I→Xbe a leafwise loop in Lxbased at xand U-covered by I. Take any point v∈dom hI, and let y=σi0(v)∈Ui0. Since the d-diameter of σi0(Ti0)is smaller than δ, according to our application of the Effros theorem, there is some φ∈Homeo(X, F)with φ(x)=yand d(c(t), φc(t))<for all t∈I. Hence the leafwise path φc ∶I→ Xis  U-covered by I. It follows that ˜ hI(v)=pi0φc(1)=pi0φ(x)=pi0(y)=v, obtaining hI(v)=v. This shows that hI=iddom hI, and therefore Hsatisfies the condition of being strongly locally free with this S. His strongly quasi-analytic because it is strongly locally free, and therefore the hypotheses of Theorem A are satisfied. In particular, the closure H is defined and generated by the set Sinduced by the above S. Now, let us sharpen the above argument to prove that His also strongly locally free, and therefore (X, F)is a G-foliated space for some local group Gby Proposition 2.2-(i). For any g∈Swith O=dom g, there is a sequence of admissible sequences, Ik=(ik,0,...,ik,αk), such that O⊂dom hIkfor all kand g=limkhIk∣Oin the compact-open topology. Thus i0∶=ik,0is independent of k. Suppose that g(u)=ufor some u∈O, which means that u′ k∶=hIk(u)→uas k→∞. So we can assume that ik,αk=i0for all k. Let 32 J.A. ´ ALVAREZ L ´ OPEZ AND R. BARRAL LIJ ´ O 7. C∞G-foliated spaces are C∞foliated homogeneous Suppose that Xis compact and C∞. Then the following result guarantees certain leafwise homogeneity. Proposition 7.1. Let Lbe the leaf of X, let Dbe a relatively compact regular domain without holonomy in L, and let c∶I→Dbe any C∞path. Then, for any open neighborhood Uof c(I)in X, there is some C∞leafwise diffeotopy φ∶X×I→Xsupported in Uwith φ(c(0),⋅)=c. Proof. Let Ebe a relatively compact open subset of Lsuch that c(I)⊂E and E⊂D∩U. By the homogeneity of L, there is a diffeotopy ψ∶L×I→L supported in Eso that ψ(⋅,0)=idXand ψ(c(0),⋅)=c. Let Σ be a local transversal of Xthrough x. By the Reeb’s stability theorem for C∞foliated spaces [4, Proposition 1.7], there is a C∞foliated embedding h∶D×Σ→X that can be identified with the identity on D×{x}≡Dand {x}×Σ≡Σ. Write h−1=(h′, h′′)∶imh→D×Σ. Take a compactly supported continuous function f∶Σ→Iwith h(E×suppf)⊂Uand f(x)=1. Then the statement is satisfied with the C∞foliated diffeotopy φ∶X×I→Xdefined by φ(x, t)=⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ h(ψ(h′(x), fh′′(x)), h′′(x)) if x∈imh xotherwise . Corollary 7.2. If there is a C∞right transverse local action of Gon X satisfying (5), then Xis C∞foliated homogeneous. Proof. Apply (5) and Proposition 7.1.  Proof Theorem C. By Theorem B, it is enough to prove “(iii) ⇒(i).” With the notation of Section 6.3, (G, G, µ)satisfies (1) because µ((Γ×µ({g}×Q))∩dom µ)=G for all g∈Gand Q∈N(G, e)with {g}×Q⊂domµ. So (T, H, χ)also satisfies (1) by Lemma 2.4, and therefore (X, F, φ)satisfies (5) by Lemma 6.8. Thus Xis C∞foliated homogeneous by Corollary 7.2  8. Examples and open problems 8.1. Molino’s description of equicontinuous suspensions. Let Tbe a compact space with a transitive left action of a compact topological group G, which is quasi-analytic in the sense that any g∈Gis the identity element e∈Gif it acts as the identity on some non-empty open set, and let H⊂G be the isotropy group at some fixed point u0∈T. Moreover let Γ ⊂Gbe a dense subgroup isomorphic to π1(M)/π1(L)for some regular covering Lof some closed connected manifold M. Thus we have a right Γ-action on Lby covering transformations, and a left Γ-action on Tdefined by the G-action. The induced diagonal Γ-action on L×T, given by (y, u)⋅γ=(y⋅γ, γ−1⋅u), is properly discontinuous and foliated, where L×Tis foliated with leaves L×{u}, for u∈T. The corresponding foliated quotient space, L×ΓT, is MOLINO’S DESCRIPTION AND FOLIATED HOMOGENEITY 33 called the suspension of the Γ-action on T, and the quotient projection is a foliated covering map L×T→L×ΓT. The element in L×ΓTdefined by any (y, u)∈L×Twill be denoted by [y, u]. Moreover the covering projection θ∶L→Minduces a fiber bundle projection ρ∶L×ΓT→M,ρ([y, u])=θ(y), with typical fiber T; in particular, L×ΓTis compact. Note that the fibers of ρare transverse to the leaves; i.e., ρ∶L×ΓT→Mis a flat bundle. Any flat bundle with compact total space is given by a suspension. Let us use the notation X≡(X, F)for L×ΓT. Let V={Vi, ζi}be an atlas of M, with ζi∶Vi→Bifor some contractible open subset Bi⊂Rn. Thus the flat bundle ρ∶X→Mis trivial over every Vi; i.e., there are homeomorphisms ψi∶Ui∶=ρ−1(Vi)→Vi×Tsuch that ρ∶Ui→Vicorresponds to the first factor projection Vi×T→Viand the leaves of F∣Uicorrespond to the fibers of the second factor projection Vi×T→T. We get an induced foliated atlas U={Ui, ξi}of X, where ξi=(ζi×idT)ψi∶Ui→Bi×T′ iwith T′ i≡T. Assuming obvious conditions on V, we get that Uis regular. Then Uinduces a representative H′of the holonomy pseudogroup of Xon T′=⊔iT′ i. For any fixed index i0, since T′ i0≡Tmeets all H′-orbits, by restricting H′to T′ i0, we get a pseudogroup Hon Tequivalent to H′, which is generated by the Γaction on T. Thus Xis minimal, equicontinuous and strongly quasi-analytic (take S=Γ to check the last two properties for H). Moreover His generated by the G-action on T, and therefore His also strongly quasi-analytic. So X satisfies the conditions of Theorem A. Fix some u0∈T≡T′ i0, and consider the associated space  T′ 0with the pseudugroup  H′ 0, and the associated representative of the Molino’s description, (G′, H′, X′ 0≡( X′ 0, F′ 0),ˆπ′ 0), constructed like in the proof of Theorem A. Then  T0∶= T′ i0,0meets all  H′ 0-orbits, obtaining that  H′ 0is equivalent to its restriction  H0∶= H′ 0∣ T0. Thus  T0={γ(g, u0)∣g∈G}has the final topology induced by the map G→ T0,g↦γ(g, u0). This map is a continuous bijection, and therefore it is a homeomorphism because Gis compact and  T0is Hausdorff. So  T0≡G, His generated by the action of Gon itself by left translations, G′is locally isomorphic to G, and ˆπ0∶ T0≡G→Tis the orbit map g↦g⋅u0. The composite ρˆπ′ 0∶ X′ 0→Mis a fiber bundle with typical fiber  T0≡G, and ( X′ 0, ρˆπ′ 0, F′ 0)is also a flat bundle. Thus there is a foliated homeomorphism of  X′ 0to  X0≡( X0, F0)∶=L×ΓG. Moreover H′≡H∶={h∈G∣h⋅u0=u0}, the right H′-action on  X′ 0corresponds to the right H-action on  X0given by [y, g]⋅h=[y, gh], and the map ˆπ′ 0∶ X′ 0→Xcorresponds to the map ˆπ0∶ X0→Xdefined by ˆπ0([y, g])=[y, g⋅u0], which is induced by the foliated map idL׈π0∶L×G→L×T. Thus (G, H,  X0,ˆπ0)is another representative of the Molino’s description, which will be used in the next examples. If Mis C∞, its C∞structure can be lifted to a C∞structure on L, which in turn can be lifted to L×T, which finally give rise to a C∞structure on Xso that the projection ρ∶X→Mis C∞and Tρ has isomorphic 34 J.A. ´ ALVAREZ L ´ OPEZ AND R. BARRAL LIJ ´ O restrictions to the fibers. This can be similarly applied to  X0, obtaining the C∞structure given by Proposition 5.1. The same procedure can be applied to any Riemannian metric on M, obtaining induced Riemannian metrics on Xand  X0so that the projections ρ∶X→Mand ˆπ0∶ X0→Xhave locally isometric restrictions to the leaves. The following result is well known. A proof is included for completeness. Proposition 8.1. The following properties are equivalent: (i) The Γ-action on Thas no fixed points. (ii) Γ∩gHg−1={e}for all g∈G. (iii) The canonical foliated projection L×T→Xrestricts to homeomorphisms between the leaves. Proof. Let us prove “(i) ⇔(ii)”. Given any γ∈Γ and u∈T, take some g∈G such that u=g⋅u0. Then γu =u⇔γg ⋅u0=g⋅u0⇔g−1γg ⋅u0=u0 ⇔g−1γg ∈H⇔γ∈Γ∩gHg−1={e}⇔γ=e . Let us prove “(i) ⇔(iii)”. For all y, y′∈Land u∈T, we have [y, u]= [y′, u]if and only if there is some γ∈Γ such that (y′, u)=(y⋅γ, γ−1⋅u), which means γ=eand y′=y. When the conditions of Proposition 8.1 are satisfied, Xis strongly locally free (in particular, it has no holonomy), and all leaves are homeomorphic to L. If moreover Mis C∞/Riemannian, then L×T→Xrestricts to diffeomorphisms/isometries between the leaves, obtaining that all leaves are diffeomorphic/isometric to L. 8.2. The map ˆπ0∶ X0→Xmay not be a principal bundle. Consider the canonical inclusion SO(2)⊂SO(3), and the canonical transitive analytic action of SO(3)on the sphere S2≡SO(3)/SO(2). We get an induced transitive quasi-analytic left action of the compact topological group G∶= SO(3)Non the compact space T∶=(S2)N. Fix u0∈S2whose isotropy group is SO(2), and let ¯u0=(u0, u0,...)∈T. The orbit map SO(3)→S2, g↦g⋅u0, is a non-trivial principal SO(2)-bundle, and therefore it has no global sections. Then, using the arguments of the first and second examples of [40, Section 1], it easily follows that the orbit map G→T,(gi)↦(gi)⋅¯u0= (gi⋅u0), has no local sections. Since Gis second countable, connected, compact and non-abelian, it contains a dense subgroup Γ isomorphic to the fundamental group of the closed oriented surface Σ2of genus 2 [12, Corollary 8.3]. Let Lbe the universal covering of Σ2, which is diffeomorphic to the plane. Consider the corresponding suspension foliated space, X= L×ΓT, which satisfies the conditions of Theorem A, and the corresponding Molino’s description (G, H,  X0,ˆπ0)constructed in Section 8.1, where  X0= L×ΓG,H=SO(2)N, the right H-action on  X0is given by [y, g]⋅h=[y, gh], and the map ˆπ0∶ X0→Xis defined by ˆπ0([y, g])=[y, g ⋅u0]. MOLINO’S DESCRIPTION AND FOLIATED HOMOGENEITY 35 Proposition 8.2. The map ˆπ0∶ X0→Xhas no local sections, and therefore it cannot be a principal H-bundle. Proof. Since ˆπ0∶ X0→Xis induced by idL׈π0∶L×G→L×T, any local section of ˆπ0with small enough domain defines a local section of ˆπ0∶G→T. But this map has no local sections.  8.3. Foliated homogeneity may not be told by the leaves. Proposition 8.3. If Xis foliated homogeneous, then it is without holonomy, and all of its leaves are homeomorphic one another. If moreover Xis C∞(respectively, compact and Riemannian), then all of its leaves are diffeomorphic (respectively, quasi-isometrically diffeomorphic) to each other. Proof. Elementary, using that there always exist leaves without holonomy in the first assertion, and using that the differentiable quasi-isometry class of the leaves is independent of the choice of the Riemannian metric on Xin the last assertion (see e.g. [6, Proposition 10.5]).  Let us exhibit an example where the reciprocal of Proposition 8.3 does not hold. To begin with, let G1and G2be second countable, connected compact topological groups, and let G=G1×G2. Assume that G1is non-abelian. Let us use the notation g=(g1, g2)for the elements of G; in particular, we use e=(e1, e2)for the identity element. Proposition 8.4. There exists a subset P⊂G×G, which is both residual and of full Haar measure, such that, for all (g, h)∈P, the subgroup ⟨g, h⟩is dense in Gand freely generated by gand h, and ⟨g, h⟩∩({e1}×G2)={e}. Proof. By [12, Proposition 8.2], there are subsets, O⊂G×Gand O1⊂ G1×G1, which are residual and of full Haar measure, such that, for all (g, h)∈Oand (a, b)∈O1, the subgroup ⟨g, h⟩(respectively, ⟨a, b⟩) is dense in G(respectively, G1) and freely generated by gand h(respectively, aand b). Then the statement is satisfied with P=O∩{(g, h)∈G×G∣(g1, h1)∈O1}. Take G2=SO(3), and consider SO(2)⊂SO(3)and S2≡SO(3)/SO(2) like in Section 8.2. By Proposition 8.4, Ghas a dense subgroup Γ freely generated by two elements such that Γ ∩({e1}×SO(3))={e}. Hence the first factor projection G1×SO(3)→G1restricts to an injection Γ →G1, and Γ does not meet any conjugate of {e1}×SO(2)in G(all of them are contained in {e1}×SO(3)). Consider the canonical left action of Gand Γ on T∶=G1×S2≡G/({e1}×SO(2)). There is a regular covering L of the closed oriented surface of genus two, Σ2, whose group of covering transformations is isomorphic to Γ. Consider the corresponding suspension foliated space, X=L×ΓT, which satisfies the conditions of Theorem A, and the corresponding Molino’s description (G, H,  X0,ˆπ0)constructed in Section 8.1, where  X0=L×ΓG,H=SO(2), the right H-action on  X0 is given by [y, g]⋅h=[y, gh], and the map ˆπ0∶ X0→Xis defined by 36 J.A. ´ ALVAREZ L ´ OPEZ AND R. BARRAL LIJ ´ O ˆπ0([y, g])=[y, g⋅u0]. We can equip Σ2with C∞and Riemannian structures, and consider the induced C∞and Riemannian structures on Xand  X0. Since H≠{e},Xis not foliated homogeneous by Theorem C (or Theorem B and Proposition 3.2). However this cannot be seen by comparing any pair of leaves since all of them are isometric to L, and Xhas no holonomy by “(ii) ⇔(iii)” in Proposition 8.1. This argument cannot produce matchbox manifolds because Proposition 8.4 requires Gto be connected to apply [12, Proposition 8.2]. Examples with totally disconnected local transversals are given in [23, Theorem 35] and [19, Theorem 10.7]. 8.4. Inverse limits of minimal Lie foliations. This example was suggested by S. Hurder. Let (X, G)be the McCord solenoid defined as the projective limit of a tower of non-trivial regular coverings between closed connected manifolds, ⋯→Mk φk ÐÐÐ→ Mk−1→⋯→M0. Let Γk=π1(Mk), and consider the induced tower of homomorphisms between finite groups, ⋯→Γ0/Γk→Γ0/Γk−1→⋯→Γ0/Γ1, whose inverse limit Kcontains a canonical dense copy of Γ0. Then (X, G) can be also described as the suspension foliated space  M0×Γ0K, where  M0 is the universal covering of M0. We get induced maps ψk∶X→Mk, whose restrictions to the leaves are covering maps. Suppose that M0is equipped with a minimal Lie G0-foliation F0, for some simply connected Lie group G0. Then every Mkcan be endowed with the minimal Lie G0-foliation Fk∶=(φ1⋯φk)∗F0. On every G-leaf M, consider the pull-back of F0by ψ0∶M→M0. These foliations on all leaves of Gcan be combined to form a foliated structure Fon X, which is a “Lie G0-subfoliated structure” of Gin an obvious sense. We can write F=ψ∗ 0F0, which equals ψ∗ kFkfor all k. Extending the notation of suspensions, we can also write (X, F)= ( M0, F0)×Γ0K, where  F0is the lift of F0. It easily follows that (X, F)is a minimal G-foliated space for G=G0×K. 8.5. Open problems. 8.5.1. Strong quasi-analyticity of H.This problem was proposed in [10]. It is really unknown to the authors if the strong quasi-analyticity of His needed in Theorem A. More precisely, assuming that His a minimal compactly generated equicontinuous strongly quasi-analytic pseudogroup, is Hstrongly quasi-analytic? If minimality is not assumed, then counterexamples can be easily given. But the minimal case seems to be an interesting open problem. Among the wild matchbox solenoilds of [33] there might be counterexamples. MOLINO’S DESCRIPTION AND FOLIATED HOMOGENEITY 37 8.5.2. Functorality, universality and uniqueness of the Molino’s description. It would be desirable to have a uniqueness of the Molino’s description stronger than Proposition 3.1, stating that not only the structures (G, Γ, H,  X0,ˆπ0)constructed in the proof of Theorem A, but also all possible structures (G, Γ, H,  X0,ˆπ0)satisfying the conditions of its statement are equivalent. This would follow by showing a universality property, which in turn would follow by exhibiting its functoriality with respect to some kind of foliated maps. Since the definition of  X0uses germs of maps in H, the functoriality of Molino’s description could be achieved by showing that foliated maps between equicontinuous foliated spaces induce morphisms between the closures of their holonomy pseudogroups. This would be an extension of the case of Riemannian foliations, solved in [9, 8]. Such functiorality, universality and uniqueness of the Molino’s description is not even proved in the Riemannian foliation case. A direct consequence would be that His finite if and only if Xis a virtually foliated homogeneous foliated space (a finite fold covering of Xis foliated homogeneous as foliated space). 8.5.3. How large is the class of inverse limits of minimal Lie foliations? Since any metrizable locally compact local group of finite topological dimension is locally isomorphic to the direct product of a Lie group and a compact zero-dimensional topological group [34, Theorem 107], it was asked by S. Hurder whether any compact minimal foliated homogeneous foliated space of finite “topological codimension” can be realized as inverse limit of minimal Lie foliations, like in Section 8.4. This would generalize the results of [15] (see also [2]), where an affirmative answer is given for homogeneous matchbox manifolds (the case of codimension zero). If this is true, using also the Molino’s description, it could be possible to prove that any equicontinuous foliated space satisfying the conditions of Theorem A is an inverse limit of Riemannian foliations. 8.5.4. Molino’s descriptions without assuming strong quasi-analyticity. This problem arises from the Molino spaces constructed by Dyer, Hurder and Lukina in [19] for equicontinuous matchbox manifolds, where strong quasianalyticity is not needed. Their Molino spaces are also foliated homogeneous, and their leaves cover the leaves of the original matchbox, but they may not be unique. Thus the following question makes sense. 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