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Leaf closures of Riemannian foliations: a survey on topological and geometric aspects of Killing foliations

A smooth foliation is Riemannian when its leaves are locally equidistant. The closures of the leaves of a Riemannian foliation on a simply connected manifold, or more generally of a Killing foliation, are described by flows of transverse Killing vector fields. This offers significant technical advantages in the study of this class of foliations, which nonetheless includes other important classes, such as those given by the orbits of isometric Lie group actions. Aiming at a broad audience, in this survey we introduce Killing foliations from the very basics, starting with a brief revision of the main objects appearing in this theory, such as pseudogroups, holonomy and basic cohomology. We then review Molino's structural theory for Riemannian foliations and present its transverse counterpart in the theory of complete pseudogroups of isometries, emphasizing the connections between these topics. We also survey some classical results and recent developments in the theory of Killing foliations. Finally, we review some topics in the theory of singular Riemannian foliations and discuss singular Killing foliations, also proposing a new approach to them via holonomy metric pseudogroups and the theory of blow-ups, which possibly opens up a new area of interest.


Introduction
A foliation on a Riemannian manifold is called Riemannian if its leaves are locally equidistant. Alternatively, the leaves of a Riemannian foliation are locally defined by fibers of a Riemannian submersion. These objects, first presented by B. Reinhart in [59], form a very relevant class of foliations, whose research has been quite active since their introduction [68,Appendix D]. As noted by G. Thorbergsson in his survey [66], in the last two decades the theory of singular Riemannian foliations started to play an important role in the theory of submanifolds and isometric actions. In addition, singular Riemannian foliations appear naturally in all noncompact spaces of non-negative curvature, having played a fundamental role in the proof of the smoothness of the metric projection onto the soul, as noted in the work of B. Wilking [69].
There is a rich structural theory for Riemannian foliations, due mainly to P. Molino, that asserts, among other results, that a complete Riemannian foliation F admits a locally constant sheaf of Lie algebras of germs of local transverse Killing vector fields C F whose action describes the dynamics of F, in the sense that for each leaf L x ∈ F one has where L x denotes the closure of L x . Using this one verifies that the partition F := {L | L ∈ F} of M is a singular foliation, meaning that it is a smooth partition into embedded submanifolds of varying dimension.
In this work we are primarily interested in the so-called Killing foliations, that is, those Riemannian foliations which are complete an whose Molino sheaf C is globally contant. In other words, for a Killing foliation F there exists transverse Killing vector fields X 1 , . . . , X d such that T F = T F ⊕ X 1 , . . . , X d . This class of foliations includes Riemannian foliations on simply-connected manifolds and foliations given by orbits of isometric Lie group actions. This motivates the study of the class of Killing foliations, since it contains important subclasses of Riemannian foliations whilst presents relevant technical advantages, in comparison to general Riemannian foliations.
The main goal of this article is to survey the classical theory of Riemannian and Killing foliations, including Molino's structural theory and the pseudogroup approach to the transverse geometry of these foliations due mostly to A. Haefliger, and present some recent developments on Killing foliations via a deformation technique. In addition, we present the basics on singular Riemannian foliations and introduce the concept of singular Killing foliations. Recent developments regarding leaf closures of singular Riemannian foliations and the solution of Molino's conjecture allow us to conclude that relevant classes of singular Riemannian foliations are Killing, for instance, infinitesimally closed foliations defined on simply connected manifolds.
This article is organized as follows. In Section 2 we introduce the basics of foliation theory and transverse geometry, including the language of pseudogroups to treat holonomy and the notion of basic cohomology. In Section 3 we define Riemannian foliations and see some examples and classical results, including the structural theory for pseudogroups of local isometries. For this end, we also briefly review the basics of sheaf theory in this section. Next, we survey Molino's structural theory for Riemannian foliations in Section 4, establishing some relations of it with the structural theory for pseudogroups of isometries. Section 5 introduces Killing foliations and presents its main examples. This section also brings a deformation technique for Killing foliations that allows one to deform such a foliation into a Riemannian foliation with all leaves closed, whilst some topological and geometric transverse properties are mantained. Sections 6 and 7 survey recent applications of these techniques, which allows one to reduce the study of the transverse geometry of these foliations to classical geometry and topology of orbifolds. We then move to the second part of this paper, consisting of singular foliations. In Section 8 we revisit the concept of singular Riemannian foliation as a natural generalization of the regular case, and introduce some of the technical machinery from this area. After that, Section 9 is dedicated to survey the recent results concerning the proof of Molino's conjecture and introduce the analog notion of the Molino sheaf in the singular setting. Finally, in Section 10 we propose the concept of singular Killing foliations and see that this class contains relevant subclasses (such as that of homogeneous Riemannian foliations), which motivates its study. This section also brings a new approach to this class of foliations via pseudogroups of local isometries on metric spaces, which seem to be an useful tool. Figure 1. A foliation is locally defined by submersions

Foliations
Let M be a smooth n-dimensional connected manifold. A regular foliation of M is a partition F of M into p-dimensional, connected, immersed submanifolds, called leaves, such that the module X(F) of smooth vector fields that are tangent to the leaves is transitive on each leaf. This means, more precisely, that for each L ∈ F and each x ∈ L one can find smooth vector fields X i whose values at x form a basis for T x L. We denote the distribution defined by the tangent spaces of the leaves by T F and the leaf containing x by L x . The number q = n − p is the codimension of F. In Section 8 we will introduce singular foliations, which drop the requirement that all leaves have the same dimension. Until then we will often omit the word "regular" when referring to regular foliations.
There are several equivalent definitions for regular foliations (see for instance [48,Section 1.2]). Here we recall the following one, which will be specially useful. A regular foliation F is equivalently defined by an open cover {U i } i∈I of M , submersions π i : U i S i , with S i ⊂ R q open, and diffeomorphisms γ ij : π j (U i ∩ U j ) π i (U i ∩ U j ) satisfying for all i, j ∈ I. The collection (U i , π i , γ ij ) is a Haefliger cocycle representing F and each U i is a simple open set for F (see Figure 1). We will assume without loss of generality that the fibers π −1 i (x) are connected, in which case they are called plaques. Plaques glue together to form immersed submanifolds, the leaves of F. Example 2.1 (Pullbacks). Let F be a foliation of M and f : N M a smooth map that is transverse to each leaf. Then f defines a foliation f * (F) on N as follows. If (U i , π i , γ ij ) is a cocycle representing F, then f * (F) is given by the cocycle (V i , π i , γ ij ), where V i = f −1 (U i ) and π i = π i • f | V i . Observe that T f * (F) = df −1 (T F) and that codim(f * (F)) = codim(F).
Example 2.2 (Homogeneous foliations). Lie group actions constitute a main source of foliations. Precisely, recall that when µ : G × M M is a smooth action, each orbit Gx is the image of an injective immersion G/G x M (see, for instance, [5,Proposition 3.14]). Thus, if we suppose that dim(G x ) is a constant function of x, it follows that the connected components of orbits of G decompose M into immersed submanifolds of constant dimension. This decomposition F is easily seen to be a foliation, because T x (Gx) = d(µ x ) e (g), so the fields V # ∈ X(M ), V ∈ g, induced by the action generate T F, showing that this is an involutive distribution.
A specific example is the following. Consider the flat torus T 2 = R 2 /Z 2 . For each λ ∈ (0, +∞), we have a smooth R-action with dim(R [x,y] ) ≡ 0. The resulting foliation is the λ-Kronecker foliation of the torus, F(λ).
Observe that when λ is irrational each leaf is dense in T 2 , while a rational λ yields closed leaves.
When a foliation F is given by the action of a Lie group we say that F is homogeneous.
whereb · [γ] denotes the image ofb by the deck transformation associated to [γ]. There is a manifold structure on M = M /π 1 (B, x 0 ) [53, p. 28] such that the orbit projection π : M M is a covering map and, if τ : M B is given by τ (π(b, t)) = ρ(b), then it is the projection of a fiber bundle with total space M , base B, fiber T and structural group h(π 1 (B, x 0 )). The action of π 1 (B, x 0 ) preserves the leaves of F, so projecting through π we obtain a foliation F on M with codim(F) = dim(T ), constructed by suspension of the homomorphism h.
As the Kronecker foliation shows, a leaf L of a foliation F need not to be closed as a subspace of the ambient manifold M . We denote the set of leaf closures by F := {L | L ∈ F}. Understanding F is part of the study of the dynamics of the foliation. In the simple case when F = F, that is, when all the leaves of F are closed, we say that F is a closed foliation. A submanifold N ⊂ M is saturated if it is a union of leaves or, equivalently, if N = π −1 (π(N )), where π : M M/F is the projection to the leaf space. We say that F is transversely compact when M/F is compact.
A foliation (M, F) is tangentially orientable if T F is orientable, and transversely orientable if its normal bundle νF := T M/T F is orientable. In this case, choices of orientations for T F and νF give, respectively, a tangential orientation and a transverse orientation for F. It is always possible to choose an orientable finite covering space M of M such that the lifted foliation F is transversely (and hence also tangentially) orientable [16,Proposition 3.5.1]. In terms of a Haefliger cocycle, F is transversely oriented if and only if there is a cocycle {(U i , π i , γ ij )} representing F that satisfies det(dγ ij ) > 0 as a function on π j (U i ∩ U j ), for all i, j ∈ I.
Let (M, F) and (N, G) be foliations. A foliate morphism between (M, F) and (N, G) is a map f : M N that sends leaves of F into leaves of G. When there is a foliate diffeomorphism f : M N (that is, F is foliate and admits a foliate inverse), the foliations F and G are often said to be congruent. In particular, we may consider F-foliate diffeomorphisms f : M M . The infinitesimal counterparts of this notion are the foliate vector fields of F, that is, vector fields in the subalgebra These are precisely the fields whose local flows send leaves to leaves. Another characterization is that X ∈ L(F) if, and only if, for each submersion π : U S locally defining F we have that X| U is π-related to some vector field X S ∈ X(S) [53,Section 2.2].
The Lie algebra L(F) also has the structure of a module, whose coefficient ring consists of the basic functions of F, that is, functions f ∈ C ∞ (M ) such that Xf = 0 for every X ∈ X(F). We denote this ring by Ω 0 (F). A smooth function is basic if and only if it is constant on each leaf and also if and only if it factors through each submersion π : U S locally defining F to a smooth function on the quotient S [53, Section 2.1].
The quotient of L(F) by the ideal X(F) yields the Lie algebra l(F) of transverse vector fields. For X ∈ L(F) we denote its induced transverse field by X ∈ l(F). Notice that each X defines a unique section of νF and that l(F) is also a Ω 0 (F)-module.
2.1. Holonomy. We start this section by recalling the language of pseudogroups. Let S be a smooth manifold. Recall that a pseudogroup H of local diffeomorphisms of S consists of a set of diffeomorphisms h : U V , where U and V are open sets of S, such that The H -orbit of x ∈ S consists of the points y ∈ S for which there is some h ∈ H satisfying h(x) = y. The quotient by the corresponding equivalence relation, endowed with the quotient topology, is the space of orbits of H , that we denote S/H . If we have two pseudogroups of local diffeomorphisms H and K of S and T , respectively, a smooth equivalence between H and K is a maximal collection Φ of diffeomorphisms from open sets of S to open sets of T such that {Dom(ϕ) | ϕ ∈ Φ} covers S, {Im(ϕ) | ϕ ∈ Φ} covers T and, for all ϕ, ψ ∈ Φ, h ∈ H and k ∈ K , we have The collection of all changes of charts of an atlas A of a smooth manifold defines a pseudogroup H A on the disjoint union of the images of the charts. If B is a compatible atlas then one has a smooth equivalence H A ∼ = H B . More generally: Example 2.4 (Orbifolds). An n-dimensional smooth orbifold is an equivalence class O = [(S, H )] of pseudogroups of local diffeomorphisms, with S an n-dimensional manifold, satisfying that |O| := S/H is Hausdorff and paracompact and each x ∈ |O| has a neighborhood U such that H | U is generated by a finite collection of diffeomorphisms of U . Orbifolds are generalizations of manifolds that appear naturally in many areas of mathematics, for instance as quotients of manifolds by properly discontinuous actions, the so-called good orbifolds. We refer to [17], [ is the holonomy pseudogroup of F associated to γ, that we denote by H γ . If δ is another Haefliger cocycle defining F then H δ is equivalent to H γ , so we can define, up to equivalence, the holonomy pseudogroup of F. We will write (S F , H F ) to denote both this equivalence class and a specific representative in it, for it seldom leads to confusion. It is clear that S F /H F is precisely the M/F of F endowed with the quotient topology. Notice also that there is an isomorphism l(F) X(S F ) H F sending X ∈ l(F) to the vector field in X(S F ) H F given, on each S i , by X S i . In fact, in general, the study of the transverse geometry of F is the study of the H F -invariant geometry of S F . B and π i : U i T is the trivial projection. Then H F is just the pseudogroup generated by h(π 1 (B, x 0 )) < Diff(T ), encoding the recurrence of the leaves on T .
The notion of fundamental group can be generalized to pseudogroups by considering homotopy classes of H -loops, that is, sequences of continuous paths c i : [t i−1 , t i ] S, for 1 ≤ i ≤ n, and elements h i ∈ H such that h i c i (t i ) = c i+1 (t i ), for 1 ≤ i ≤ n − 1, and c 1 (0) = h n c n (1) = x. We refer to [62] and [17, Section 2.2] for details. In particular, for the holonomy pseudogroup H F a foliation (M, F) this furnishes an invariant π 1 (F, x), the transverse fundamental group of F, which captures information of both the topology of F and the holonomy of the leaves. Its isomorphism class does not depend on the Haefliger cocycle representing F nor on the base point, when M is connected (in this case we omit it, denoting simply π 1 (F)).
between small enough neighborhoods of x = π 1 (x) and y = π m (y). If we identify S γ with a total transversal i S i for F containing x and y, this becomes the "sliding along the leaves" notion from [53], Section 1.7 (see Figure 2). Let us denote the germ of γ imi 1 at x by h c . This germ actually depends only on the ∂[0, 1]-relative homotopy class of c [16, Proposition 2.3.2], hence, if we consider in particular the holonomy group of L at x, that is, the group L is a loop}, we have a surjective homomorphism h : π 1 (L, x) Hol x (L). 6 As the isomorphism class of Hol x (L) does not depend on x, we often omit x in this notation. In particular, we can say that L is a leaf without holonomy (or a generic leaf ) when Hol(L) = 0. It follows immediately from the surjectivity of h that simply-connected leaves are without holonomy. Also, it can be shown that leaves without holonomy are generic, in the sense that Suppose Hol x (L) is finite and identify it with a subgroup of Diff(S), where S is a small local transversal of F passing through x. With this in mind we can state the famous Reeb Stability Theorem as follows (see [48,Theorem 2.9] or [16, Theorems 2.4.3 and Theorem 3.1.5]). Theorem 2.6 (Generalized local Reeb stability). Let F be a smooth foliation with a compact leaf L x . If Hol x (L) is finite then there is a saturated tubular neighborhood pr : Tub(L x ) L x restricted to which F is congruent to the foliation given by the suspension of h : In particular, for every y ∈ Tub(L) the projection pr : L y L x is a finitely-sheeted covering map, the number of sheets being the index |Hol x (L x ) : Hol y (L y )|. This indicates that leaf holonomy plays the same role of the stabilizer in the case of group actions. In fact, using Theorem 2.6 one proves the following [48,Theorem 2.15].
Proposition 2.7 (Leaf space of closed foliations). Let F be a q-codimensional foliation of M whose every leaf is compact and with finite holonomy. Then M/F has a canonical q-dimensional orbifold structure, which we will denote by M//F in order to distinguish it from the topological space M/F. Relative to this structure, the local group of a leaf in M/F is its holonomy group. Moreover π 1 (F) coincides with the orbifold fundamental group π orb 1 (M//F). 2.2. Basic Cohomology. Let (M, F) be a smooth foliation. A covariant tensor field ξ on M is F-basic if ξ(X 1 , . . . , X i ) = 0, whenever some X i ∈ X(F), and L X ξ = 0 for all X ∈ X(F). In particular, we say that a differential form ω ∈ Ω i (M ) is basic when it is basic as a tensor field. By Cartan's formula, ω is basic if, and only if, i X ω = 0 and i X (dω) = 0 for all X ∈ X(F). These are the differential forms that project to differential forms in the local quotients S and are invariant by the holonomy pseudogroup of F [53, Proposition 2.3]. We denote the Ω 0 (F)module of basic i-forms of F by Ω i (F). Then is the ∧-graded algebra of basic forms of F.
By definition, Ω(F) is closed under the exterior derivative, so we can consider the complex The cohomology groups of this complex are the basic cohomology groups of F, that we denote by H i (F). A foliate map f : (M, F) (N, G) pulls basic forms on N back to basic forms on M and hence induces a linear map f * : H i (G) H i (F). When the dimensions dim(H i (F)) are all finite (see Example 3.6), we define the basic Euler characteristic of F as the alternate sum In analogy with the manifold case, we say that b i (F) := dim(H i (F)) are the basic Betti numbers of F. When F is the trivial foliation by points we recover the classical Euler characteristic and Betti numbers of M .
Since we have an identification between F-basic forms and H F -invariant forms on S F and an identification between differential forms on an orbifold O and H O -invariant forms on U O , Proposition 2.7 gives us the following. Proposition 2.8. Let (M, F) be a foliation such that every leaf is compact and with finite holonomy. Then the projection π : M M//F induces an isomorphism of differential complexes π * : Ω(M//F) Ω(F). In particular, H(F) ∼ = H dR (M//F). Example 2.4). Following [36, Section 3.2], we define a smooth foliation F of O as a smooth foliation of U A which is invariant by H A . The atlas can be chosen so that on each U i the foliation is given by a surjective submersion with connected fibers onto a manifold S i . The holonomy pseudogroup of F, therefore, will be generated by the local diffeomorphisms of the disjoint union i∈I S i that are projections of elements of H A . All notions defined so far for foliations on manifolds therefore extend to foliations on orbifolds.

Riemannian foliations
Let F be a smooth foliation of M . A transverse metric for F is a symmetric, positive, Fbasic (2, 0)-tensor field g T on M . In this case (M, F, g T ) is called a Riemannian foliation. A Riemannian metric (in the usual sense) g on M is called bundle-like for F if for any open set U and any Y, Z ∈ L(F| U ) perpendicular to the leaves we have g(Y, Z) ∈ Ω 0 (F| U ). In this case, setting defines a transverse metric for F, where we write X = X + X ⊥ with respect to the decompo-  where λ i ∈ (0, 1] and z i ∈ S n ⊂ C k+1 . We will call these foliations generalized Hopf fibrations, since we get the usual Hopf fibration when λ i = 1 for each i. In particular, such an action correspond to a closed Riemannian 1-foliation F of S n precisely when all λ i are rational, say λ i = p i /q i . Notice that in this case we can equivalently assume that λ i ∈ N, by changing the parameter t to lcm(q 1 , . . . , q k )t, hence S n //F is a weighted projective space CP k [λ 0 , . . . , λ k ]. Let us visualize these foliations in the case of the 3-dimensional sphere, that is, for k = 1. Consider the action of T 2 = S 1 × S 1 on S 3 by (t 0 , t 1 ) · (z 0 , z 1 ) = (t 0 z 0 , t 1 z 1 ). This action has two singular orbits, T 2 (1, 0) and T 2 (0, 1), that are diffeomorphic to S 1 . The other orbits are tori and coincide with the distance tubes of the two singular orbits. The 1-dimensional Riemannian foliations of S 3 , up to congruence, can be identified with the 1-dimensional Lie subalgebras of R 2 ∼ = lie(T 2 ) via the induced action of the corresponding 1-parameter subgroup. They restrict to Kronecker foliations on each regular T 2 -orbit (see Figure 3).  Bundle-like metrics can be characterized in terms of its geodesics: It follows from this result that the leaves of a Riemannian foliation are locally equidistant. Contrarily to the classical case of Riemannian metrics on manifolds, not every smooth foliation admits a transverse metric so that it becomes a Riemannian foliation. This will become more apparent when we study Molino's structural theorem in Section 4, but we can already conclude that from the fact that the basic cohomology of Riemannian foliations on compact manifolds have finite dimension (Theorem 3.7 below), which is not true for smooth foliations in general:

Example 3.6 ([24]). Consider
and its induced diffeomorphism A : Then we have an homomorphism Z ∼ = π 1 (S 1 ) Diff(T 2 ) given by n A n . Ghys shows in [24] that the 1-dimensional foliation F given on the torus bundle T 3 A S 1 by the suspension of this homomorphism has dim(H 2 (F)) = ∞ (see Examples 2.3 and 2.5). In fact, it can be verified by direct calculations that an A-invariant 1-form on T 2 (corresponding to a basic 1-form) is of the type f (x)dx, thus closed, while an A-invariant 2-form is of the type g(x)dx ∧ dy. Therefore H 2 (F) must be infinite-dimensional. As remarked in [26,Proposition 3.11], the hypothesis that M is compact can be relaxed to F being transversely compact, provided that F is a complete Riemannian foliation, in the following sense. It follows that χ(F) is always defined for transversely compact (i.e., such that M/F is compact) complete Riemannian foliations. We mention the following transverse analogue of the Bonnet-Myers Theorem due to J. Hebda: Theorem 3.9 ([38, Theorem 1]). Let F be a complete Riemannian foliation satisfying Ric F ≥ c > 0. Then F is transversely compact and H 1 (F) ∼ = 0.
Basic cohomology of Riemannian foliations can be studied via the basic Laplacian ∆ B . Let F be a transversely oriented Riemannian foliation of a compact oriented manifold M endowed with a bundle-like metric g. Consider the scalar product ·, · B in Ω i (F) given by the restriction of the usual scalar product in Ω i (M ) (see, e.g., [57, Section 2 of Chapter 7]). The basic laplacian is the operator ∆ B : Ω i (F) Ω i (F) given by ∆ B = dδ + δd, where δ is the formal adjoint of d with respect to ·, · B . We denote by H i (F) the space of basic harmonic i-forms, that is, basic i-forms α satisfying ∆ B α = 0. For a thorough introduction to this objects, we refer to [68,Chapter 7].
There is a basic version of Hodge's decomposition theorem for ∆ B that gives an orthogonal decomposition (see [68,Theorem 7.22]) and so provides an isomorphism (see also [68,Theorem 7.51]) This leads to duality theorems for the basic cohomology. Poincaré duality in its expected form, however, is only available for the so-called taut foliations: a Riemannian foliation F of M is taut it there exists a Riemannian metric on M with respect to which every leaf of F is a minimal submanifold. Tautness is also characterized in [44,Theorem 6.4] by the vanishing of a degree 1 cohomology class. In particular, if Ric F ≥ c > 0, one concludes from Theorem 3.9 that F is taut.
3.1. Complete Pseudogroups of Local Isometries. As we mentioned earlier, the transverse information of a Riemannian foliation corresponds to the holonomy-invariant information on a total transversal, so, in considering transverse geometry, one can focus on the later. In this section we survey this point of view, pioneered by A. Haefliger, focusing mainly on the study the closures of the orbits of a complete pseudogroup of isometries. The main references are [35], [61] and [62].
It follows directly from the definition of a Riemannian foliation (M, F, g T ) that the transverse metric g T projects to Riemannian metrics on the local quotients S i of a Haefliger cocycle {(U i , π i , γ ij )} defining F (see [53,Section 3.2], and also [48, Remark 2.7(2)]). Therefore the holonomy pseudogroup H F becomes a pseudogroup of local isometries of S F . Moreover, by choosing a bundle-like metric on M , the submersions π i become Riemannian submersions. Definition 3.11 (Complete pseudogroups). We say that a pseudogroup of local isometries (H , S) is complete when, given x, y ∈ S, there exists neighborhoods U x and V y such that every germ of an element of H with source in U and target in V is the germ of an element of H defined on the whole of U . This property is invariant by differentiable equivalences [62, p. 278]. It is also independent of the concept of completeness in the sense of Riemannian manifolds, as the following example shows. . Suppose H is a pseudogroup of local diffeomorphisms of S whose equivalence class represents an orbifold O. One can always choose a Riemannian metric on O, which corresponds to an H -invariant Riemannian metric on S. Then O is not necessarily complete as a Riemannian orbifold, but we claim that H is a complete pseudogroup of local isometries. In fact, for every point x ∈ S one can find an H -invariant neighborhood U x. Hence, if x, y ∈ S are in the same H -orbit then every germ of an element of H with source and target in U is the germ of an element defined on the whole of U (here we take V = U y to be the neighborhood of y used in the definition of completeness). On the other hand, if x and y are in different orbits, then since S/H is Hausdorff one can separate the orbits H x and H y by two disjoint open neighborhoods U ⊃ H x and V ⊃ H y. Therefore there are no germs of elements of H with source in U and target in V .
The example below establishes the connection between complete Riemannian foliations and complete pseudogroups of local isometries. A proof can be seen in [62, p. 281]. It follows essentially from Proposition 3.5. Let H be a complete pseudogroup of local isometries. Its closure H is defined as the pseudogroup on S whose elements are locally the limits, in the C 1 topology, of elements of H .
We say that H is closed when H = H .

3.2.
A brief interlude on sheaves. Before we continue it will be convenient to recall the notion of sheaves, which are tools for working with locally defined data on topological spaces. A presheaf P on a topological space (X, τ ) consists of an assignment of a set P(U ), to each U ∈ τ , and a restriction map res U V : One often is interested in local data (the sets P(U )) that have additional structure, such as algebraic operations. In this case one requires that the restriction maps preserve the additional structure. This leads to the definition of presheaves of groups, rings and so on. For example, if each U is assigned to a (real) Lie algebra P(U ) and each res U V is a Lie algebra homomorphism, then P is a presheaf of Lie algebras. R, together with the usual restriction of functions is a presheaf of rings C ∞ M . Given a presheaf P on (X, τ ) and x ∈ X, let U x be the collection of open sets that contain x. For U 1 , U 2 ∈ U x , declare s 1 ∈ P(U 1 ) and s 2 ∈ P(U 2 ) to be equivalent if there exists V ∈ U x such that V ⊂ U 1 ∩U 2 and res U 1 V (s 1 ) = res U 2 V (s 2 ). The equivalence class of s ∈ P(U ) is the germ of s at x, denoted by res U x (s) or simply by [s] x . The set P x of germs at x is called the stalk of P at x. Notice that the stalks of a presheaf of structured sets (say groups or Lie algebras) inherit that structure in a natural way.
A sheaf on a topological space on (X, τ ) is a presheaf S on (X, τ ) such that, for any U ∈ τ and any open covering A sheaf of groups (or rings, Lie algebras etc.) is just a presheaf of groups (or rings, Lie algebras etc.) that is a sheaf in the above sense.
Example 3.17. It is not difficult to check that, for a smooth manifold M , the presheaf C ∞ M is a sheaf of rings. Now let π : E M a smooth vector bundle. The presheaf that assigns to each open set U the space of smooth local sections of E over U , with the usual restriction maps, is a sheaf of C ∞ M -modules. Sections of a sheaf S can be realized as (usual) sections of its étalé space. In fact, given a presheaf P on (X, τ ), its étalé space is the space endowed with the (in general non-Hausdorff) topology whose basis is given by the sets of the form There is a canonical projection which is a local homeomorphism, by construction. The presheaf Γ(Et(P)) of local sections of π P (that is, continuous maps s : U Et(P) with π P • s = id U ) is a sheaf, called the sheafification of P, or the sheaf of germs of sections of P. When P is already a sheaf, it is isomorphic to Γ(Et(P)).
Example 3.18. A presheaf of sets (or groups, Lie algebras, etc) P on X said to be constant when P(U ) = Z for all U ∈ τ , where Z is a fixed set (or group, Lie algebra, etc). In this case the stalks are all Z, hence we can identify Et(P) ∼ = X × Z. Under this identification, the étalé topology corresponds to the product topology on X × Z, where Z is given the discrete topology, and π P : X × Z X is just the projection on the first factor. More generally, a presheaf P is locally constant when every x ∈ X admits an open neighborhood U x where P| U is constant. In this case the identification Et(P| U ) ∼ = U × Z is a local trivialization of P.
Sheaves can be "transported" through continuous maps, as follows. Suppose f : X Y is continuous and R is a sheaf on X. Then we define the direct image of R by f as the sheaf . On the other hand, if we have a sheaf S on Y then we can also obtain a sheaf f −1 S on X, called the inverse image of S by f , which consists of the sheaf of germs of sections of the It is instructive to try to understand f −1 S (U ) in terms of the values of S on open sets of Y , although this is a little involved since f (U ) is not necessarily an open set. To circumvent this, we need to generalize the notion of germ, as follows. For any subset A ⊂ Y , let U 1 and U 2 be open neighborhoods of A. We will say that two sections s 1 ∈ S (U 1 ) and Infinitesimal sheaf of complete pseudogroups. There is a structural theorem for complete pseudogroups of local isometries, due to E. Salem [61], which describes the closures of the orbits of such a pseudogroup as orbits of a sheaf of Lie algebras on it. This result can also be seen as a generalization of Myers-Steenrod Theorem. Let H be a pseudogroup of diffeomorphisms of S. Consider the sheaf which to each open set U ⊂ S associates the space C H (U ) of vector fields X on U such that, for all x ∈ U there exists a neighborhood V x x and ε > 0 such that exp(tX) is defined on V x , when |t| < ε, and exp(tX) ∈ H . Definition 3.19 (Sheaf of infinitesimal transformations). With the notation above, the sheaf C H is called the sheaf of infinitesimal transformations of H .
For a complete pseudogroup of local isometries H of S, the sheaf C H is a locally constant sheaf of Lie algebras of germs of Killing vector fields on S (see [61,Proposition]). Thus, if S/H is connected, all stalks of C H are therefore isomorphic to a Lie algebra g −1 . The Lie algebra g will be called the structural Lie algebra of H . . If H is generated by a closed subgroup G of isometries of a Riemannian manifold M , then C H is the sheaf whose sections are the restrictions of the fundamental Killing vector fields of the action, that is, elements in the image of the infinitesimal action of the Lie algebra g X(M ) of G. Hence C H is isomorphic to the constant sheaf with stalk g −1 on S.
A complete pseudogroup of local isometries H is a Lie pseudogroup when any element of H that is close enough to the identity is of the form exp(X), for a local section X of C H close to 0. Hence C H plays an analogous role as the Lie algebra of a Lie group. As a corollary, it follows that the orbits of H are closed submanifolds of S, since they are given by the orbits of the sheaf C H . In other words, the closure of an H -orbit is described, on a small open set U ∈ S where C H is constant, by the flows the Killing vector fields in C H (U ).
In the case of the holonomy pseudogroup of a complete Riemannian foliation, we can use Theorem 3.21 to similarly describe the closures of the leaves as orbits of a sheaf, since we have M/F = S/H by Proposition 3.14. To make this more precise, we need the following definition.
Definition 3.22 (Transverse Killing vector field). A field X ∈ X(M ) is a Killing vector field for g T if L X g T = 0. These fields form a Lie subalgebra of L(F) and there is, thus, a corresponding Lie algebra of transverse Killing vector fields, that we will denote by iso(F, g T ). We will omit the transverse metric when it is clear from the context, writing just iso(F).
If (U i , π i ) is a Haefliger cocycle defining F, the elements of iso(F) are precisely the transverse fields that project to H F -invariant Killing vector fields on S. The inverse images π −1 i (C H ) of the infinitesimal sheaf C H of H hence patch together on M to form a sheaf C F of Lie algebras of germs of transverse Killing fields (see [62, §3.4], also [36, Remark at p. 711]).

Molino Theory
Molino theory consists of a structural theory for Riemannian foliations developed by P. Molino and others in the decade of 1980. In this section we summarize it, following mostly the brief presentations in [26,Section 4.1] and [67,Section 3.2]. A thorough introduction can be found in [53].
Let (M, F, g T ) be a q-codimensional Riemannian foliation. Recall that g T induces a Riemannian metric π * (g T ) on the local quotient S of each foliation chart (U, π). Consider the pullback to U of the Levi-Civita connection on S determined this metric. By uniqueness, the pullbacks obtained this way glue together to a well-defined connection ∇ B on T M , the canonical basic Riemannian connection, where "basic" means that the local foliate fields in L(F| U ) are parallel along the leaves with respect to ∇ B . Note that ∇ B induces a covariant derivative on l(F| U ), that we also denote by ∇ B .
Let π : M F M be the principal O(q)-bundle of F-transverse orthonormal frames 1 , which we call the Molino bundle of F. The normal bundle νF is associated to M F and so the basic Riemannian connection ∇ B on νF induces a connection form ω F on M F . This connection form in turn defines an O(q)-invariant horizontal distribution H := ker(ω F ) on M F that allows us to horizontally lift the leaves of F, defining this way an O(q)-invariant foliation F of M F . By construction, ω F is F -basic, that is, i X (ω F ) = 0 and L X ω F = 0 for all X ∈ X(F ), so we may regard it as a map ω F : νF so(q).
A practical way to think of F is the following: if x = π (x ) and y = π (y ) then y belongs to the leaf L x ∈ F if and only if the orthonormal frame y of ν y F is the parallel transport of the frame x , with respect to ∇ B , along some smooth path in L x from x to y.
We can also define a transverse metric for F so that it becomes a Riemannian foliation. We lift the transverse metric g T on νF to an O(q)-invariant metric on the O(q)-invariant transverse horizontal distribution νH := H/T F . The pullback (π ) * (g T ) coincides with the pullback of the standard scalar product on R q by the fundamental 1-form θ F : νF R q defined by (see [53, p. 70 and p. 148 By the discussion above, the pullback of the sum of an arbitrary (which is unique up to scalar λ) bi-invariant scalar product on so(q) with the standard scalar product on R q by ω F ⊕ θ F yields an O(q)-invariant F -transverse metric (g T ) with respect to which F is a Riemannian foliation. Moreover, π : M F M becomes a transversely Riemannian submersion, that is, dπ is surjective and restricts to an isometry dπ : (νH) x ν x F for each x ∈ M F . We now fix λ by requiring that the fibers of π satisfy vol((π ) −1 (x)) = 1.
The advantage of lifting F to F is that the latter admits a global transverse parallelism, that is, νF is parallelizable by fields in l(F ) [53, p. 82 and p. 148]. If we assume that F is complete, then those fields admit complete representatives 2 in L(F ) [26, Section 4.1]. Now, from the theory of transversely parallelizable foliations, it follows that the partition F of M F is a simple foliation, that is, W := M F /F is a manifold and F is given by the fibers of a locally trivial fibration b : We now study the restriction of F to a leaf closure through this construction. Fix L ∈ F , denote J = L , consider the foliation (J, F | J ) and define g := l(F | J ). The restriction of F to the closure of a different leaf is isomorphic to (J, F | J ), so g is an algebraic invariant of F. Definition 4.1 (Structural algebra). The Lie algebra g is the structural algebra of F. We will always denote dim(g) by d.
The foliation F | J is a complete g-Lie foliation in the terminology of E. Fedida [21], that is, it admits a complete transverse parallelism {Z 1 , . . . , Z d } such that the Lie algebra it spans is g. Equivalently, for a real Lie algebra g, a complete g-Lie foliation F on a manifold J is given Fedida's work establishes, by a classical argument of C. Ehresmann, that complete g-Lie foliations are developable, that is, they lift to simple foliations on some covering space (see [53,Theorem 4.1]). In fact, let G be the unique simply connected Lie group with Lie algebra g and consider J × G with projections pr 1 and pr 2 on the first and second factors, respectively. Let L α := {X ∈ L(F) | α(X) is constant} be the subalgebra of foliate vector fields whose corresponding transverse fields are in g. Via the identifications T (J × G) ∼ = T J ⊕ T G and g ∼ = T e G, define the lift of X ∈ L α by which is a G-invariant vector field on J × G, with respect to the natural left action of G. This lifting is R-linear and commutes with the Lie brackets, so the lift of L α is a Lie algebra of left invariant vector fields which defines a left invariant integrable distribution ∆ of rank dim(J) on J × G. Let J be a leaf of the corresponding foliation. J is a covering map and pr 2 : J G is a locally trivial fibration. Moreover, the foliation pr * 1 (F) on J agrees with the simple foliation defined by the fibers of pr 2 . We see that a complete g-Lie foliation admits a Haefliger cocycle (U i , π i : U i G, γ ij ) such that the transitions γ ij are restrictions of left translations on G. Moreover, since π 2 is Gequivariant, the holonomy pseudogroup of F is equivalent to the pseudogroup generated by the induced action of the group Γ of deck transformations of π 1 on G.
Let us now return to a complete Riemannian foliation (M, F, g T ). Consider on M F the sheaf of Lie algebras C F that, to an open set U ⊂ M F , associates the Lie algebra C F (U ) of the transverse fields in U that commute with all the global fields in l(F ). The orbits of C F are the closures of the leaves of F [53, Theorem 4.3']. Each field in C F (U ) is the natural lift of a local F-transverse Killing vector field on π (U ) [53,Proposition 3.4], which in turn is the lift of a section of the sheaf of infinitesimal transformations of H F . So we conclude that the direct image π * (C F ) coincides with the Molino sheaf C F (recall Definition 3.23). In fact, this is how C F was originally defined by Molino 5 .
All stalks of C F are isomorphic to the Lie algebra g −1 opposed to g [53, Proposition 4.4]. As we already stated, the main motivation for the study of C F is that its orbits describe the closures of the leaves of F. In other word, this means that Let us summarize the properties seen in this section in the following theorem, known as Molino's structural theorem. (v) There is a locally constant sheaf C F of Lie algebras of germs of transverse Killing vector fields whose stalks are g −1 and whose orbits are the closures of the leaves of F.

Killing foliations
From now on we will be mostly interested in the subclass of complete Riemannian foliations consisting of those foliations F for which C F is globally constant. Such foliations are called Killing foliations, following the terminology of W. Mozgawa in [54]. In other words, if F is a Killing foliation then there exists X 1 , . . . , X d ∈ iso(F) such that From the point of view of the holonomy pseudogroup H F , the foliation F is Killing if, and only if, its sheaf of infinitesimal transformations C H F has an H F -invariant global trivialization. In other words, C H F admits a global trivialization which is invariant by the local isometries in H F . We say, accordingly, that a complete pseudogroup of local isometries is a Killing pseudogroup when its sheaf of infinitesimal transformations is (invariantly) globally trivial.
It is also easy to see that a complete Riemannian foliation is a Killing foliation if and only if C F is globally constant, and in this case C F (M F ) is the center of l(F ). Hence C F (M ) is central in l(F), but not necessarily its full center. It follows that the structural algebra of a Killing foliation is Abelian, because For this reason, when F is Killing we will often denote g F simply by a, provided there is no risk of confusion.   ). This is an example of a Killing foliation which is not homogeneous and whose ambient manifold is not simply connected. It is constructed by a slight modification of the suspension method (Examples 2.3 and 3.3). Consider T = T 2 × T 2 . For A ∈ SL 2 (Z), if v is an eigenvector of A, the foliation given by lines in R 2 that are parallel to v projects to a Kronecker foliation F v of T 2 (see Example 2.2). We choose A so that F v is not closed, e.g. by requiring that tr(A) > 2. Seeing this torus T 2 as the second factor of T , the product foliation of the trivial foliation {T 2 } on the first factor with F v gives us a codimension 1 foliation F T of T with dense leaves. Consider the diffeomorphism Φ A := id × A : T T , where A : T 2 T 2 is the diffeomorphism determined by A. The suspension of the homomorphism π 1 (S 1 ) Diff(T ) given by n Φ n A furnishes us a fiber bundle τ : M S 1 with fiber T and structural group Φ A . Here we are not interested in the foliation given by this suspension, but rather the foliation F induced fiberwise on M by F T , which is well defined since F T is invariant by Φ A . One sees immediately that F is Riemannian and its leaf closures are the fibers of τ .
Since Φ A acts trivially on l(F T ), and F T is clearly transversely parallelizable, it follows that C F is globaly trivial, that is F is a Killing foliation. Notice, however, that M is not simply connected. It remains to verify only that F is also not homogeneous. In fact, if this were the case, F would be given by the orbits of a connected Lie subgroup H < Iso(M, g), with respect to some Riemannian metric g on M . Then F, given by the fibers of τ , would coincide with the orbits of H, hence one could conclude that τ is associated to a principal H-bundle E S 1 . Since H is connected, E should be trivial, hence also M S 1 would be trivial. But this does not happen by construction: the map Φ * A : H(T ) H(T ) induced by the generator Φ A of its structural group on the homology of the fibers is non-trivial, hence M S 1 is not topologically trivial.
In [54] W. Mozgawa establishes some implications of Molino's structural theorems in the case of Killing foliations: Theorem 5.4 (Mozgawa's Theorem [54,Théorème]). Let F be a q-codimensional Killing foliation of a closed manifold M of dimension n = p + q. If r + p = min L∈F dim(L) then: (i) There exists r commuting transverse Killing vector fields X 1 , . . . , X 1 ∈ iso(F) which are everywhere linearly independent, and (ii) The orbits of the Lie algebra span(X 1 , . . . , X 1 ) define a Riemannian foliation F of M of codimension q − r which has at least one closed leaf and satisfies F = F.
In particular, it follows easily from this theorem that, if χ(M ) = 0, then every Killing foliation F on M has at least one closed leaf. More recently it was shown in [18] that a much stronger conclusion holds when M is compact: if χ(M ) = 0 then every leaf of F is closed (see Theorem 6.4).

5.1.
Transverse structure of Killing foliations. The transverse structure of a Killing foliation coincides with that of an (Abelian) homogeneous foliation on an orbifold, as established by A. Haefliger and E. Salem in [36]. More precisely, by comparing the local models of the transverse structure of a Killing foliation on a neighborhood of a leaf closure and the local model of an orbit of a torus action on an orbifold, the authors obtain the following.
where Γ is the corresponding subgroup of translations of R d in A 2 helps clarifying the relation between T N and F. Note, in particular, that N ≥ d.
The existence of Υ follows non-trivially from the theory of classifying spaces of pseudogroups, developed by Haefliger in [33]: the classifying space of H F is a space BH F with a foliation BF such that the holonomy covering of each leaf is contractible and H F ∼ = H BF . As in the classical case of classifying spaces in homotopy theory, there is a map Υ : M BH F , whose homotopy class is unique up to homotopy along the leaves, which is transverse to BF and such that F = Υ * (BF). The point is that A 1 A 3 associates the class of F to a canonical representative Example 5.6. In the simple case of an irrational generalized Hopf fibration F of S 3 (see Eaxample 3.2), the construction of (O, T N , H, µ) is trivial: O = S 3 with the action of T N = T 2 by restriction of the multiplication on C 2 , and H is the subgroup determined by the R-action that defines the foliation. To also illustrate item (ii) of Theorem 5.5, recall that the restriction of F to the closure of a generic leaf is an irrational Kronecker foliation F(λ) (see Example 2.2). Notice moreover that F(λ) is a Lie R-foliation, so in view of Theorem 4.2, H F (λ) is equivalent to the pseudogroup generated by the group Γ of translations of R induced, via projection along the lifted foliation on the universal covering R 2 , by the action of π 1 (T 2 ) by deck transformations. Notice that in fact we have rank(Γ) = 2 = rank(π 1 (T 2 )), since the generators of π 1 (T 2 ) project to rationally independent translations of R (FAZER FIGURA).
More generally, for a Killing foliation with compact leaf closures F, if L ∈ F is a generic leaf, the authors establish in [ for i = 0, 1. Here we will simply say that F t is a deformation of F 0 into F 1 .
For a Riemannian foliation F on a simply connected, closed manifold M , E. Ghys showed in [23,Théorème 3.3] that is possible to deform F into a closed foliation G, in such a way that the deformation respects F, that is, it occurs within the closures of the leaves of F. As remarked by the authors in [36], Theorem 5.5 generalizes this result: for a Killing foliation F on a closed manifold M , consider a corresponding orbifold (O, T N , H, µ) and the map Υ : M O such that F = Υ * (F H ). Let h be the Lie algebra of H and slightly perturbate it into a Lie subalgebra k < Lie(T N ) ∼ = R N , with dim(k) = dim(h), such that its corresponding Lie subgroup K < T N is closed. If k is close enough to h (as points in the Grassmannian Gr dim h (Lie(T N ))), it is possible to choose a smooth path h(t) connecting h to k such that for each t the action µ| H(t) of the corresponding Lie subgroup H(t) is locally free and the induced foliation F H(t) remains transverse to Υ. Then F t := Υ * (F H(t) ) defines a deformation of F = F 0 into G = F 1 . It is possible to prove that H Ft is equivalent to H F H(t) for each t. Moreover, since K is closed, G is a closed foliation and, by construction, the deformation respects F.
The "transverse homogeneous" nature of this deformation allows one to preserve some geometric properties of F in G. This was investigated in [18]: One can then use the Riemannian geometry and topology of M//G to study F. In the next sections we will summarize some applications of this technique.

Transverse topology of Killing foliations
Let us begin by studying the basic Euler characteristic of a Riemannian foliation F on a closed manifold M . Recall that, by Theorem 3.7, χ(F) is always well defined. In [10], the authors establish a transverse version of Hopf's index theorem: The concepts involved, such as critical leaf closures, nondegeneracy and the index ind J (X), are generalizations of the classical ones. Since the analog of a classical critical point is a leaf closure, it is natural to expect that its topology must be taken into account in Theorem 6.1. 20 This is encoded in the factor χ(J, F, Or J (X)), the alternate sum of the cohomology groups of the complex of F| J -basic forms with values in the orientation line bundle of X at J (for more details, see [10, Section 3]). Theorem 6.1 can be used to show that χ(F) localizes to the strata of closed leaves: Theorem 6.2 ([18, Theorem D]). If F is a Killing foliation of a compact manifold M , then In particular, if F has no closed leaves, then χ(F) = 0.
In fact, in [18, Theorem 7.1] the authors prove something stronger: if X ∈ iso(F), then χ(F) = χ(F| Zero(X) ). This is in analogy to the classical localization of the Euler characteristic of a Riemannian manifold to the zero set of a Killing vector field (see, e.g. [57, Theorem 40]) or, alternatively, to the fixed point set of a torus action. Theorem 6.2 then follows by choosing a Killing vector field X ∈ C F (M ) such that F| Zero(X) = Σ dim F /F (see [18,Proposition 3.4]).
Combining Theorems 6.2 and 5.7, if G is a closed foliation approximating F, then we have that this observation proves the following: In particular, for the closed foliation G = F 1 , Theorem 6.3 reduces questions about χ(F) to questions about χ(F) = χ(M//G) = χ(|M//G|). An interesting application comes from the results in [33], where Haefliger studies the classifying spaces BH f . It follows from his work that, similarly to the case of fiber bundles, χ(M ) has the product property (1) χ(M ) = χ(L)χ(G), for L ∈ G a generic leaf (see [33, Corollaire 3.1.5]). Now assume F is non-closed but has a closed leaf L, and, by regular deformations, choose a sequence G i of closed foliations approaching F. Then L ∈ G i for each i, since the deformations preserve F. In particular, using Theorem 2.6 we can rewrite equation (1) as [18,Lemma 4.3]). This would violate equation 2 unless χ(M ) = 0. As we mentioned before, it follows by Theorem 5.4 that every Killing foliation of a manifold M with χ(M ) = 0 has a closed leaf, so from the discussion above one concludes the following. Notice that Theorem 6.4 is in fact a slight improvement of [18, Theorem 9.1] (which is stated for a simply-connected M ), but the proof is essentially the same. By lifting to the universal covering, one has the following corollary for Riemannian foliations: We conclude from Example 6.6 that the basic cohomology groups H(F) are not preserved by deformations. In the next section we will see, however, that there is a cohomological invariant, namely, basic equivariant cohomology, that is preserved. This will, in particular, provide sufficient conditions for the basic Betti numbers to be preserved as well.
6.1. Equivariant basic cohomology. When a group G acts on a space M , there is a cohomology theory that captures information on both the topological space X and the action of G on it, simultaneously generalizing ordinary cohomology theory and group cohomology. It is called equivariant cohomology, and defined as the singular cohomology of the Borel construction: where EG is a contractible space on which G acts freely (e.g., the total space of the universal G-bundle EG BG). The motivation for this is that the diagonal action of G on EG × M is free, so the quotient is a well-behaved space (in contrast to M/G). A remarkable feature of equivariant cohomology, with no counterpart in classic cohomology, is that the non-torsion part of the module structure of H T (M ), for a torus space M , can be recovered from the fixed point set M T . This is known as Borel localization. We will see a transverse counterpart of this result below. We refer to [46] and [27] for more detailed introductions to the classical equivariant cohomology theory, and to [9] and [32] for thorough treatments of this topic.
It turns out that, when M is a G-manifold (and provided G is compact and connected), there is another way to compute H G (M, R), via the de Rham complex Ω(M ) and the Lie algebra g. It was obtained by H. Cartan (actually before Borel's definition of H G (M, R)) and the fact that it computes H G (M, R) is considered as the equivariant analog of the classical de Rham theorem (see, e.g., [32, Theorem 2.5.1]). We are interested in Cartan's model for equivariant cohomology because its algebraic nature makes it readily generalizable to our transverse setting.
Recall that a differential g -algebra is a Z-graded-commutative differential algebra (A, d) endowed, for each X ∈ g, with derivations L X and ι X , of degree 0 and −1, respectively, satisfying If A and B are g -algebras, an algebra morphism f : A B is a morphism of g -algebras if it commutes with d, L X and ι X . Example 6.7. An infinitesimal action of a Lie algebra g on an orbifold O is a Lie algebra homomorphism µ : g X(O) such that the map g × O T O given by (X, x) µ(X) x is smooth. A differential g -algebra structure on (Ω(O), d) is then given by the usual Lie derivative L X = L µ(X) and interior product ι X := ι µ(X) .
If we have a Lie group G action µ on O, then there is an induced infinitesimal action of its Lie algebra: each X ∈ g induces the fundamental vector field X # ∈ O given by Consider also the coadjoint action of a Lie algebra g on its dual algebra g ∨ given, for X, Z ∈ g and Y ∨ ∈ g ∨ , by ad ∨ X Y ∨ (Z) = Y ∨ (−[X, Z]). It extends naturally to the symmetric algebra S(g ∨ ) over g ∨ . The space C g (A) := (S(g ∨ ) ⊗ A) g of those elements on S(g ∨ ) ⊗ A which are g-invariant, with respect to the coadjoint action and the derivation L on the first and second factors, respectively, is the Cartan complex of A. Notice that an element ω ∈ C g (A) can be identified with a polynomial map ω : g A. Under this identification, g-invariancy of ω as an element of C g (A) becomes g-equivariancy of ω as a polynomial map g A: ω(ad X Y ) = −L X ω(Y ). The equivariant differential d g of the Cartan complex is then defined as In order for it to be a derivation of degree 1, the grading on C g (A) is defined by The Cartan model for the equivariant cohomology of A is .
Example 6.8. In the case of a G-orbifold O, the S(g ∨ ) g -module structure S(g ∨ ) g C g (O) coincides with the cohomology map induced by the constant map O { * }.
A g -algebra A is said to be equivariantly formal if S(g ∨ ) g ⊗ H(A) as S(g ∨ ) g -modules. Equivalently, A is equivariantly formal when H a (A) is a free S(a ∨ )-module. There are several relevant classes of equivariantly formal algebras. For instance, for a manifold M with a torus action, and A = Ω(M ) with the induced t -structure, A is equivariantly formal when H odd (M ) = 0, or when M is symplectic and the torus action is Hamiltonian.
We now go back to the case of a foliation F on an orbifold O. A transverse infinitesimal action of a Lie algebra g on F is a Lie algebra homomorphism is some foliate field representing µ(X). It induces a g -algebra structure on Ω(F), with d being the usual exterior derivative and the derivations L X and ι X defined as L X ω := L X ω and ι X ω := ι X ω (see [26,Proposition 3.12]). We can therefore define the g-equivariant basic cohomology of F as the g-equivariant cohomology of Ω(F), which we will denote H g (F) := H g (Ω(F)) = H(C g (Ω(F), d g )).
Now consider a Killing foliation F on M . In this case we have a natural transverse infinitesimal action of its structural algebra a, given by the isomorphism a ∼ = C F (M ). Notice that the fixed point set M a = {x ∈ M | a x = a} is precisely the union of the closed leaves of F, since aF = F. These two facts, that the infinitesimal a-action is canonical and that aF = F, makes the study of H a (F) very relevant.
The equivariant basic cohomology H a (F) was first introduced in [26], where the authors show that, in analogy to classical equivariant cohomology, it satisfies a Borel-type localization. Before we state their result it will be useful to recall the notion of R-module localization from commutative algebra. Given an R-module A and a multiplicative subset S ⊂ R we define the localization of A at S by S −1 A = (A × S)/ ∼, where (a, s) ∼ (a , s ) if there is r ∈ S such that r(s a + sa ) = 0. We can think of an equivalence class (a, s) as fraction a/s. Notice S −1 A is an S −1 R-module with the usual operation rules for fractions. Additionally, map of R-modules ϕ : A B induces a map of S −1 R-modules S −1 ϕ : S −1 A S −1 B by a/s ϕ(a)/s.
This result was recently generalized to transverse actions of Abelian Lie algebras on transversely compact Riemannian foliations in [43]. I follows from Theorem 6.9 that the kernel of For the next result, we recall that the transverse action of a on F is equivariantly formal when Ω(F) is an equivariantly formal a -algebra. In this case we also say that F is equivariantly formal. A transversely orientable Killing foliation F is equivariantly formal, for example, when some of the following conditions hold (see [26]): (iii) F admits a basic Morse-Bott function whose critical set is equal to M a , The dimension dim(H(F)) of basic cohomology can be studied via equivariant cohomology, providing another consequence of Theorem 6.9: The behavior of equivariant basic cohomology under regular deformations was studied in [19]. Recall the construction of F t as a pullback Υ * (F H(t) ) from section 5.2. Notice there is a transverse action of t/h(t) on F H(t) for each t. All those Lie algebras are isomorphic to a, although in a non-canonical way. We define an a-action on F H(t) (and thus on F t , since H (F t ) ∼ = H (F H(t) )), by passing through an isomorphism t/h(t) a, which amounts to identifying a with a subalgebra of t complementary to each h(t), that by abuse we will also denote by a < t. It is now possible, therefore, to consider the a-equivariant basic cohomology of F t , that is, H a (F t ). Of course, one will be specially interested in H a (G), for which one has In particular, for t = 1 we have H a (F) ∼ = H a (G) ∼ = H T d (M//G), as rings, for a closed foliation G arbitrarily close to F, thus reducing the study of H a (F) to equivariant cohomology of torus actions on orbifolds. Moreover, the authors show in [19,Proposition 6.2] that equivariant formality is preserved by regular deformations, that is, if F is equivariantly formal, then each F t is equivariantly formal with respect to the transverse a-action on given in Proposition 6.12. Hence, in this case Recall that the Poincaré series of an N-graded vector space V is the formal power series P V (s) = ∞ k=0 (dim V k )s k (provided dim V k finite for each k), which has the following product property: P V ⊗W (s) = P V (s) P W (s). Passing to the Poincaré series in equation (3) and canceling out P S(a ∨ ) (s) on both sides yields P H(F ) (s) = P H(Ft) (s). This proves the following: Theorem 6.14 ([19, Theorem B]). If F is equivariantly formal and F t is a regular deformation, then b i (F t ) is constant on t, for each i.

Transverse geometry of Killing foliations
Many techniques from classical Riemannian geometry can be used in the study of the transverse geometry of Riemannian foliations, as our brief survey on Section 2.3 already illustrates, and many classical theorems admit a transverse generalization. We also cite here the following result by G. Oshikiri: This is obtained by studying zeros of transverse Killing fields via classic techniques. The existance of a closed leaf in item (i) corresponds to the existence of a zero for a transverse Killing vector field, and thus is a transverse analog of classical Berger's theorem on zeros of Killing vector fields (see, e.g., [57,Theorem 38]). Notice that if sec M > 0, with respect to a bundle-like metric for F, then sec F > 0, since by O'Neil's formula [55] applied to a Riemannian submersion locally defining F one has for X, Y ∈ L(F). Also in positive transverse curvature, Hebda's Theorem 3.9 is obtained essentially by the study of focal points of leaves over horizontal geodesics. In the case of nonpositive transverse curvature, Hebda proves that leaves have no focal points, which then leads to the following. 7.1. Transverse geometry via deformations. An inherent difficulty often encountered in these aforementioned transverse generalizations of classical theorems from Riemannian geometry is that the leaf space of a Riemannian foliation has, in general, an ill-behaved topology which in many cases renders direct generalizations of "local-to-global" theorems impossible. For Killing foliations this difficulty can in some cases be circumvented by the deformation technique we presented in Section 5.2, since some aspects of transverse geometry are preserved by regular deformations. In this section we will see several applications of this approach, that appeared in [18] and [19]. For instance, by combining the deformation method with the Synge-Weinstein theorem for orbifolds [71,Theorem 2.3.5] one can relax the hypothesis on Theorem 7.1: There is also an application involving Bochner's theorem on Killing vector fields in the context of negative Ricci curvature [57,Theorem 36]. This result adapts directly to orbifolds [19,Theorem 2.5] and, via deformations, implies the nonexistence of transverse Killing fields for a Ricci negatively curved Killing foliation, which is therefore closed (cf. Theorem 7.2). For the next result, recall the notion of π 1 (F) from Section 2.1. Recall also that the growth function # of a finitely generated group Γ = g 1 , . . . , g k is the function that associates to j ∈ N the number of distinct elements in Γ which can be written as words with at most j letters in the alphabet {g 1 , . . . , g k , g −1 1 , . . . , g −1 k }. Then Γ is said to have exponential growth if #(j) ≥ α j for some α > 1 (this property is independent of the set of generators [47,Lemma 1]). Milnor's theorem establishes that the fundamental group of a negatively curved compact manifold has exponential growth [47, Theorem 2]. Milnor's proof of this result adapts to orbifolds [19,Theorem 2.6], and then the deformation method can be used to show the following. . Let F be a Killing foliation on a compact manifold M such that sec F < 0. Then F is closed and π 1 (F) grows exponentially. In particular, π 1 (M ) grows exponentially.
One should compare Theorem 7.5 with [38,Theorem 3], which implies that a compact manifold whose fundamental group is nilpotent does not admit a Riemannian foliation with sec F < 0, recalling Gromov's theorem that states that a finitely generated group has polynomial growth if and only if it has a nilpotent subgroup with finite index [30,Main Theorem]. Another result by Gromov establishes an upper bound for the sum of Betti numbers of negatively curved manifolds, in terms of their dimension and volume [30, p. 12]. An analogous bound holds for orbifolds, as shown by I. Samet in [63,Theorem 1.1]. Combining it with with the fact that a negatively curved Killing foliation is closed, by Theorem 7.5, we get: Corollary 7.6. There exists a constant C = C(q) such that, for any Killing foliation F on a compact manifold M with sec F < 0, say −k 2 ≤ sec F < 0, one has The classical Singe's theorem also has an orbifold version, proved by D. Yeroshkin  Recall that the symmetry rank symrank(M ) of a Riemannian manifold M is the rank of its isometry group, that is, the dimension of a maximal torus in Iso(M ). It was proven by K. Grove and C. Searle in [31]   In the transverse setting, Theorem 6.3 guarantees that one can study the basic Euler characteristic by the deformation method. A generalization of the Püttmann-Searle theorem for orbifolds was proven in [18,Theorem 8.9], from which a transverse version for Killing foliations follows by deformation: Theorem 7.9 ([18, Theorem E]). Let F be a q-codimensional transversely orientable Killing foliation of a compact manifold M . If q is even, sec F > 0 and symrank(F) ≥ q/4 − 1, then χ(F) > 0.
Finally, Theorem 6.14 shows that the basic Betti numbers of Killing foliations can also be studied via deformations, provided the transverse action of the structural algebra is equivariantly formal. For instance, a theorem by Gromov establishes the existence of a constant C = C(n) that bounds the total sum of Betti numbers of any positively curved Riemannian manifold of dimension n [29, §0.2A]. An analogous result holds for orbifolds, as it follows by [42,Theorem 1]. Thus, by deformations, one obtains the following transverse generalization: Theorem 7.10 ([19, Theorem E]). There exists a constant C = C(q) such that every qcodimensional Killing foliation F of a compact manifold M with sec F > 0 and whose transverse action of the structural algebra a is equivariantly formal satisfies

Singular Riemannian foliations
In this section we will briefly present singular Riemannian foliations and survey some classical and recent results about them. The notion of singular foliation generalizes that of regular foliations by allowing the dimensions of the leaves to vary. More precisely, given an n-dimensional connected manifold M , a singular foliation of M is a partition F of M into connected, immersed submanifolds, called leaves, such that the module X(F) of smooth vector fields that are tangent to the leaves is transitive on each leaf. This means, as in the regular case, that for each L ∈ F and each x ∈ L one can find smooth vector fields X i whose values at x form a basis for T x L. We maintain most of the notation from regular foliations, e.g. we denote the distribution of varying rank defined by the tangent spaces of the leaves by T F and the leaf containing x by L x . The algebra of foliate vector fields can also be defined similarly, as L(F) = {X ∈ X(M ) | [X, X(F)] ⊂ X(F)}, and consists of those fields whose flows take leaves to leaves. The transverse vector fields are the elements of l(F) := L(F)/X(F). The dimension of F is defined as dim(F) = max L∈F dim(L). . One easily verifies that T x Hx = µ(h)| x , that is, the space generated by the fundamental vector fields of the H-action at x is the tangent space of the orbit Hx at x. This shows that the partition F H of M into the connected components of the orbits of H is a singular foliation. In analogy with the regular case, such a foliation is an homogeneous singular foliation. One also verifies that F H = F H e , where H e < H is the connected component of the identity, so supposing that H is connected usually does not affect the study of F H and has the advantage that in this case the leaves (which are connected by definition) coincide with the orbits.
Singular Riemannian foliations are defined by generalizing Reinhart's characterization of bundle-like metrics (Proposition 3.5): if M can be endowed with a Riemannian metric g such that every geodesic which is perpendicular to a leaf of F remains perpendicular to all leaves it intersects, then we say that F is a singular Riemannian foliation and that g is adapted to F. Any partition of M into submanifolds (not necessarily a smooth singular foliation) having this property is called a transnormal system on (M, g), following the terminology of [12]. We can say, hence, that a singular Riemannian foliation F of M is a singular foliation of M which is also a transnormal system with respect to some Riemannian metric.
For a leaf L ∈ F we denote the normal space at x ∈ L by ν x L = (T x L) ⊥ . It is clear from Proposition 3.5 that every regular Riemannian foliation F is a singular Riemannian foliation. Homogeneous singular Riemannian foliations form another very significant class: Proposition 8.2 ([53, Section 6.1]). Let (M, g) be a Riemannian manifold on which a Lie group H acts by isometries. Then g is an adapted metric for F H , which is thus a singular Riemannian foliation.
It follows from Molino's structural theorem (Theorem 4.3) that the closure F of a complete regular Riemannian foliation is a singular foliation. One has, in fact, the following. . Let (M, F) be a complete (regular) Riemannian foliation and g be a bundle-like metric. Then F is a singular Riemannian foliation to which g is adapted.
In Section 9 we will see that a similar result holds for a complete singular Riemannian foliation F: the partition F of M into the closures of leaves of F is again a singular Riemannian foliation.
One defines basic cohomology in complete analogy with the regular case: for a singular Riemannian foliation F, a differential form ω ∈ Ω i (M ) is basic if ι X ω = 0 and L X ω = 0 for all X ∈ X(F). The d-subcomplex of F-basic forms will be denoted by Ω(F). It is a Z-graded differential algebra with respect to the usual exterior derivative and wedge product. The basic cohomology of F is the cohomology H(F) of (Ω(F), d). 8.1. Slice foliation, homothetic lemma and canonical stratification. In this section we review some basic technical notions that will be useful. Let L ∈ F be a leaf of a complete singular Riemannian foliation of M , and consider a tubular neighborhood U := Tub ε (P ) of radius ε > 0 of a connected, relatively compact, open subset P ⊂ L. That is, U is the image of B P ε := {V ∈ νP | V < ε} by the normal exponential map exp ⊥ : νL M , where ε is taken small enough so that exp ⊥ | B P ε is a diffeomorphism onto U . There is an orthogonal projection π P : U P . By decreasing ε and shrinking P if necessary, we can further assume that U is a distinguished tubular neighborhood, i.e., that it also satisfies the following: (i) L y is transverse to the slice S x := π −1 P (x) = exp ⊥ x (B ε (0)), for each y ∈ U , x = π P (y), and (ii) P is a leaf of a (regular) simple subfoliation of F| given by the fibers of a submersion ρ : U π −1 P (x). The connected component P y of L y ∩Tub ε (P ) containing y is a plaque through y. Condition (ii) is a natural generalization of the definition of regular foliations, and it is possible to check that the existence of distinguished tubular neighborhoods is in fact equivalent to the transitivity of X(F) in each leaf, in definition of singular foliations. Definition 8.5 (Slice foliation). With the notation above, we define the slice foliation at x as the foliation F| Sx of S x given by the intersections P y ∩ S x , for y ∈ Tub ε (P ).
Given a distinguished tubular neighborhood Tub ε (P ), if ε 1 , ε 2 = λε 1 ∈ (0, ε) one can define the homothetic transformation h λ : Tub ε 1 (P ) exp ⊥ (V ) − exp ⊥ (λV ) ∈ Tub ε 2 (P ). Lemma 8.6 actually holds more generally when L is, instead of a leaf, a submanifold which is saturated by leaves of F, all of them of the same dimension, and the definition on U is adapted accordingly. This result is fundamental for the theory of singular Riemannian foliations. It is used to prove, for instance, that the union Σ r of all leaves of F of dimension r, called a stratum, is an embedded submanifold [53,Proposition 6.3]. This provides a stratification M = r Σ r of M such that the restriction F r := F| Σr is a regular foliation, for each r. The stratum of the leaves of maximal dimension is the regular stratum of F, which we also denote by Σ reg = Σ dim(F ) , and all other strata are called singular. The union Σ sing of all singular strata is the singular locus of F. We will also often denote the most singular stratum by Σ min , called the minimal stratum. Using Lemma 8.6 one proves moreover that: (i) Each Σ r is transversely totally geodesic, meaning that a geodesic which is perpendicular to a leaf L ∈ Σ r and tangent to Σ r remains within Σ r and is, in particular, a geodesic of Σ r with respect to the restriction of the metric g. (ii) Thus g r := g| Σr is a bundle-like metric for F r , which is hence a (regular) Riemannian foliation. The transverse metric it induces will be denoted by g T r . (iii) If L ⊂ Σ r then L ⊂ Σ r [53, Lemma 6.4]. Furthermore, each Σ r is obviously saturated, so Lemma 8.6 can also be applied for N = Σ r , from what one concludes: (iv) All singular strata have codimension at least 2, so Σ reg is an open, dense submanifold of M .
Definition 8.7 (Transverse Killing vector fields). We say that a transverse field X ∈ l(F) is a transverse Killing vector field of F if its restriction to each stratum Σ r is a transverse vector field for (F r , g T r ) (see Item (ii) above). The algebra of F-transverse Killing vector fields will be denoted by iso(F).

8.2.
Orbit-like, infinitesimally closed and linearized foliations. As we previously saw, for each x ∈ M we have a slice foliation F| Sx of a slice S x . Its pullback by the exponential map is a singular Riemannian foliation of B ε (0) ⊂ T x S x with respect to g x and thus, by Lemma 8.6, can be extended via homotheties to a singular Riemannian foliation F x on the whole of T x S x , called the infinitesimal foliation at x. Notice that if F is regular, then F x is the trivial foliation of T x S x by points.  The next example turns out to be very relevant in geometry (see [11] and [69]). Example 8.10 (Holonomy foliations). Examples of orbit-like foliations can be constructed as follows. Suppose L is a Riemannian manifold, and E is an Euclidean vector bundle over L, with inner product , x on each fiber E x , and suppose ∇ E is a metric connection on E, that is, it satisfies X ξ, η = ∇ E X ξ, η + ξ, ∇ E X η . Then ∇ E induces a Riemannian metric g E on E, the connection (Sasakian) metric, and a parallel transport on E given as follows: for X ∈ E x and a curve γ : [0, 1] L with γ(0) = x, there exists a unique lift X(t), t ∈ [0, 1] with X(0) = X such that ∇ E γ (t) X(t) = 0 for every t ∈ [0, 1]. We define the holonomy foliation F E on E by declaring two vectors X, Y ∈ E to be in the same leaf if they can be connected to one another via a composition of parallel transports with respect to ∇ E . This defines a singular Riemannian foliation on E for which g E is adapted. For a point x along the zero section L, the infinitesimal foliation F x coincides with the homogeneous foliation given by the orbits of the holonomy group Hol x of the connection ∇ E acting by isometries on the fiber E x . Similarly, at a point X ∈ E x the infinitesimal foliation is given by the orbits in ν X L X of the stabilizer H X ⊂ H x of X. Therefore F E is infinitesimally homogeneous. In addition if the connected component of Hol x is compact, then F E is orbit-like. This happens for example if E = T L or if E = ν(L) when L is an embedded submanifold of an Euclidean space (see [11]).
We end this section with a technical construction which will be needed later. Let U = Tub ε (P ) be a distinguished tubular neighborhood, for P contained in a saturated submanifold N ⊂ Σ r . If X ∈ X(F| U ) is given, we can produce another vector field X on U given by , which is called the linearization of X with respect to P . It is smooth, invariant under homothetic transformations and coincides with X along P [50, Proposition 13]. The module X(F| U ) of linearized vector fields spans a foliation F of U , in the sense that the leaves of F are the orbits of the pseudogroup of local diffeomorphisms H (U, F) generated by the flows of the vector fields in X(F| U ) . Definition 8.11 (Linearized foliation). The foliation F of U is called the linearization of F with respect to P , or just the linearized foliation when F and P are clear from the context. The metric g in general is not adapted to the linearization F , but it is possible to construct a new metric g on U turning F into a singular Riemannian foliation (see [7,Section 7] for details. Now let S x = π −1 P (x) be an F-slice and, in analogy with the definition of the slice foliation F| Sx , consider the partition F | Sx given by the connected components of L ∩ S x as L ranges between the leaves of F . By construction, the pullback exp * x (F | Sx ) is invariant by rescalings, so we can also extend it to a foliation (F ) x of the whole T x S x . One proves that (F ) x coincides with the linearization of (F x , g x ) with respect to the origin in T x S x [53, §6.4], that is (F ) x = (F x ) , so it is safe to denote both of them by F x . The next result is the main reason why we are interested in this object. . The foliation F is the maximal infinitesimally homogenous subfoliation of F| U , for a tubular neighborhood U of a leaf closure J = L. Moreover, for each x ∈ U the foliation F x is given by the connected components of the orbits of the Lie group O(F x ) of linear isometries of (T x S x , g x ) sending each leaf of F x to itself.

Molino's conjecture and its proof
In this section we address the question of whether the closure F of a singular Riemannian foliation is again a singular Riemannian foliation. Molino made this conjecture in the 1980's, and it remained open until 2017, when it was proved positive in [7]. Before we present the results in [7], let us see what can be achieved by applying the structural theorem for regular foliations (Theorem 4.3) to each stratum of a singular Riemannian foliation. 9.1. Molino sheaf of a singular Riemannian foliation. Consider a complete singular Riemannian foliation (M, F). As we saw in item (ii) above, the restriction of a singular Riemannian foliation F to each stratum Σ r is a regular Riemannian foliation. Although F r is not necessarily complete, one can still apply Theorem 4.3 to it because its holonomy pseudogroup is complete. It therefore follows that its leaf closures are submanifolds of Σ r , and in particular the leaf closures in F are submanifolds of M . Moreover, the distance between two leaf closures is locally constant, since this is true for the leaves themselves, hence the partition F is a transnormal system on M .
Each F r has a locally constant Molino sheaf C r := C Fr of germs of local transverse Killing vector fields that describes the closure F r . Consider, in particular, the Molino sheaf C reg of the restriction F reg to the regular stratum. The opposite Lie algebra of its stalk is called the structural Lie algebra of F, and denoted simply by g. The motivation for this is that C reg extends continuously to a locally constant sheaf C F on M , the Molino sheaf of F, with stalk g −1 [53,Lemma 6.5]. In fact, let us briefly present how this extension is obtained. Suppose Σ r is the stratum of the singular leaves of maximal dimension and let P ⊂ Σ r be an open, relatively compact, simply connected subset. If codim(Σ r ) = 2, then by Lemma 8.6 one concludes that the restriction of F to the boundary of a tubular neighborhood Tub ε (P ) is the pullback π * P (F r ). So, C reg coincides with π −1 P (C r ) on Tub ε (P ), hence the result. Now, if codim(Σ r ) > 2, then Tub ε (P ) \ P is a simply connected open subset of Σ reg , on which C reg is therefore constant and thus extends to Tub ε (P ) by continuity. The extension of C reg to the other strata is then done similarly. Furthermore, as the above example suggests, it is possible to prove that each sheaf C r is a quotient of C F [53, Proposition 6.8]. In particular, the structural algebra g r of F r is a quotient of g, for each r.
9.2. Blow ups and desingularization. Let us review a useful technical tool for singular Riemannian foliations that allows one to "desingularize" a singular Riemannian foliation F on a compact manifold M by constructing from it another compact Riemannian manifold (M B , g B ), a regular foliation F B , and a foliate smooth map ρ : M B M with good geometric properties. For instance, ρ restricts to a foliate diffeomorphism outside Σ B := ρ −1 (Σ sing ) and to an isometry outside a narrow open neighborhood of Σ B . This technique, inspired by the blowup methods for resolution of singularities in algebraic geometry, appeared in [52] for closures of regular Riemannian foliations and was generalized to arbitrary singular Riemannian foliations on compact manifolds in [4].
The foliation (M B , F B , g B ) is obtained by successively blowing it up along the most singular strata. It is instructive to review this process in more detail. Denote by Σ := Σ min its minimal stratum and by U := Tub ε (Σ) a tubular neighborhood of Σ. Σ is the canonical projection. 32 (iii) There exists a singular foliation F on U whose leaves have dimension strictly greater then those of F and so thatρ : ( U \ Σ, F) (U \ Σ, F) is a foliate diffeomorphism. In addition, if F is homogeneous then the leaves of F are also homogenous. (iv) There exists a Riemannian metricĝ on U adapted to F.
Let us briefly recall the construction of the metric in item (iv). Consider the smooth distribution S on U given by S y = T y S x , where S x is a slice of L x at x with respect to the original metric and denote y = exp(X), for X ∈ ν x Σ. Recall that there exists a metricg so that the normal space P to S y is tangent to the leaf L y and so that (U, F, g) has the same transverse geometry as (U, F,g), i.e., the distance between the plaques is the same regardless which metric we use. Then we have the decomposition • P y is orthogonal to S y , with respect tog, • S s y ⊂ S y is tangent to the spheres exp x (νΣ ∩ B X (0)), • S r y is the line generated by d dt exp x (tX)| t=1 , and • S c y is the orthogonal complement of S s y ⊕ S r y in S y . We now define a metricĝ on U \ Σ which is adapted to F. Let f : (0, r) R be a smooth function so that f (t) = r 2 t for 0 < t < r 8 and f (t) = 1 for r 4 < t < r 2 , and set Notice thatĝ is adapted to F, because f ( exp −1 (x) ) is constant along L x and henceĝ is basic on each stratum.
Since the distribution S, i.e., the normal distribution to P with respect tog, can be deformed to the normal distribution to P with respect to g without changing the transverse metric, we can extend the metricĝ on Tub r/4 (Σ) to a new metricĝ on U so thatĝ exp(rX/4) =g exp(rX/4) and g exp(tX) = g exp(tX) , for r/2 < t and X = 1. The pullback (ρ) * ĝ defines a smooth metric on U so that F turns into a singular Riemannian foliation on U . Finally we can extend the metriĉ g on U to a metric on the connected sum such that F is a singular Riemannian foliation on M .
We can now define by induction M 1 := M and M k := M k−1 , with blow-up projection ρ k : M k M k−1 and blow-up foliation F k := F k−1 . We further define (M B , F B , g B ) as the last blow-up space in this process (which eventually ends since dim(F) is finite) and ρ : M B M as ρ = ρ n • · · · • ρ 1 .
Although in this survey we are interested in understanding singular Riemannian foliations with non-closed leaves, we recall here the following result that illustrates further geometric properties of blow-up's. M that is induces an ε-isometry between the leaf spaces, that is, if x and y are points in M B then |d(L ρ(x) , L ρ(y) ) − d(L B x , L B y )| < ε. In particular the metric space M/F is a Gromov-Hausdorff limit of a sequence of Riemannian orbifolds.
We will be specially interested in the following property. Proposition 9.3. Let F be an infinitesimally closed singular Riemannian foliation on a compact Riemannian manifold M . Then every local F-transverse Killing vector field X admits a lift to a local F B -transverse Killing vector field X B , in the sense that the flows of X and X B satisfy ρ • ϕ B = ϕ • ρ.
Remark 9.4. In the general case where F is not necessarily infinitesimally compact one has a similar conclusion if in addition one supposes that the local flows of the transverse isometries are contained in the closure of linearized holonomies, a concept that we present below. 9.3. The proof of Molino's conjecture. In [8] the authors establish Molino's conjecture for orbit-like foliations using Theorem 9.5. We already saw that F is a transnormal system, so it remained to show that any vector X x ∈ ν x L x ∩ T x L x can be extended to a smooth vector field X ∈ X(F). Notice that this is a local problem since we can find an extension X in a neighborhood of x and then use a partition of unity to extend it by 0 to a vector field in X(F). The main ingredient in the proof of Molino's conjecture in [7] is the following smooth lifting of (metric) isometries between leaf spaces. To comment on the proof we will need a generalization of the notion of slice, which will also be useful later in the study of the dynamical behavior of a singular Riemannian foliation F. Let J be an F-saturated manifold contained in some stratum Σ r and let S ⊂ U a slice of the restriction F| J . Moreover, consider the denote the restriction (νJ)| S of the normal bundle νJ to S. Definition 9.6 (Reduced space). We define the reduced space of F along S as the manifold N := exp(ν ε J)| S ), where (ν ε J)| T = {ξ ∈ (νJ)| S | ξ < ε} and ε > 0 is small enough so that exp is a diffeomorphism onto N.
With the footpoint projection, the reduced space is a fiber bundle p T : N T whose fibers contain the leaves of the foliation F N defined by the intersections of F with N. Furthermore, one can endow N with a metricg which is adapted to F N and preserves the transverse metric of F [8, Proposition 2.20].
The point of this construction is that when F is bundle-like, the foliation F N is homogeneous, given by the orbits of a compact Lie group [8,Corollary 2.25]. Hence one can apply (a slight generalization of) Theorem 9.5 to continuous flows of isometries on N/F N . Having this, it is then a matter of finding such a flow that corresponds to a given X x ∈ ν x L x ∩ T x L x . The authors accomplish that by first generalizing the notion of (regular) holonomy pseudogroup by obtaining, for N a reduced space along a slice, a pseudogroup of local metric isometries H (F N ) acting on N/F N and capturing the recurrence of the leaves on N (we will introduce a similar construction in Section 10.1). Now consider the desingularization (N B , F B N ) of F N . Since F B N is regular, Theorem 3.21 implies that H F B N is a Lie pseudogroup, so lifting the local projection of X x ∈ ν x L x ∩ T x L x to a vector in N B , which is tangent to the leaf closures, one concludes that it extends to a local Killing vector field. But there is a bijection between local isometries in H F B N and local isometries in the closure H (F N ) (in the compact-open topology) [8,Lemma 4.2], so the authors obtain the desired continuous flow and hence a smooth vector field on N that can then be extended to the desired vector field on M extending X x .
Having Molino's conjecture for orbit-like foliations, the conclusion for an arbitrary singular Riemannian foliation is obtained by using the linearization F . More precisely, on a local neighborhood a leaf L ∈ F the authors show in [7] that there is an orbit-like foliation F , obtained from F by taking the "local closure" of the leaves of F (see details in [7, §6] [7,Theorem]). Let F be a complete singular Riemannian foliation of M and let g be an adapted metric. Then (i) The closure F is a singular Riemannian foliation with adapted metric g, (ii) There exists a locally constant sheaf of Lie algebras C F on M which induces C reg on Σ reg and whose restriction to a singular stratum Σ r admits C r as a quotient sheaf.
Unlike the regular case, however, it is not possible to conclude from what we have seen so far that C F is a sheaf of Lie algebras of germs of transverse Killing vector fields, since the extension of C reg to C F is continuous: we do not know whether the extensions of sections admit smooth representatives. In fact, this would imply Molino's conjecture, since in this case for a small neighborhood U the (smooth) fields in C F (U ) would be transitive on the closures of the leaves. For this reason, we state the following: Conjecture 9.9 (Strong Molino conjecture). The sheaf C F is a sheaf of Lie algebras of germs of transverse Killing vector fields.
Molino proposes this conjecture in [53, p. 215]. It does not follow directly from the already mentioned proof of Molino's conjecture that appears in [7], but from the results in [8] that we saw in Section 9.3 one can conclude that it is true for orbit-like foliations. In fact we will see in Section 10.1 that the machinery developed in [7] and [8] can be used to conclude that the strong Molino conjecture holds for infinitesimally closed foliations.

Singular Killing foliations
In this section we propose a definition for singular Killing foliations. On the one hand, as discussed in Section 9, we do not know whether in general the Molino sheaf C F is a sheaf of germs of transverse Killing vector fields -the strong Molino conjecture. On the other hand, the fact that C F is indeed a globally trivial sheaf of germs of transverse Killing fields when F is a regular Killing foliation is of fundamental relevance for this class of foliations. So we will assume this a priori in our generalization of Killing foliations to the singular setting. Notice that the structural algebra of a singular Killing foliation F is Abelian, since it is the structural algebra of the (regular) Killing foliation F reg . We will therefore follow our notation of the regular case and denote it by a. We have an isomorphism a = C F (M ). Also in analogy with the regular case, if M is simply connected then C F is automatically globally constant. Thus a singular Riemannian foliation of a simply connected manifold is a singular Killing foliation, provided is satisfies the strong Molino conjecture. In Section 10.1 we will prove that this is the case for the class of infinitesimally closed Riemannian foliations. We also have the following: Example 10.2 (Homogeneous singular Riemannian foliations are Killing). As we saw in Example 9.1, if F is homogeneous, given by the orbits of H < Iso(M ), then C F is induced by the fundamental Killing vector fields of the action of H < Iso(M ), hence F is a singular Killing foliation.
It seems therefore relevant to study this class of foliations and investigate to what extent the results concerning regular Killing foliations that we saw in Sections 6 and 7 generalize to the singular setting. This is intended to a forthcoming paper. For now we point out, for instance, the following. Proof. The proof is analogous to that of the regular case [26,Proposition 3.12]. The transverse infinitesimal action of a is given by the isomorphism a ∼ = C F (M ) < l(F), so we can identify a ≡ C F (M ). For each X ∈ a we define the derivations L X := LX and ι X := ιX, wherẽ X ∈ L(F) represents X. Notice that these operators are well defined, since we are restricted to forms on Ω(F), and inherit the needed a -algebra relations from Ω(M ).
Therefore F possesses a natural equivariant basic cohomology, its a-equivariant basic cohomology of F, defined as the equivariant cohomology of the associated Cartan complex C a (Ω(F)) (see Section 6.1): H a (F) := H a (Ω(F)) = H(C a (Ω(F), d a )).
10.1. Linearized holonomy pseudogroups and Molino sheaf. In the previous sections we saw how the closure of the holonomy pseudogroup induces a locally constant sheaf whose local pullbacks to M glue together to the Molino sheaf, which describes the leaf closures of a Riemannian foliation (and in particular of a Killing foliation). It is natural to ask whether we can follow a similar procedure for singular Riemannian foliations and for singular Killing foliations. In this section we start to approach this question. More precisely, we will associate to an infinitesimally closed singular Riemannian foliation F of a compact Riemannian manifold M a pseudogroup of isometries H (called the linearized holonomy pseudogroup) acting on a disjoint union of metric spaces, generalizing the usual concept of the pseudogroup of holonomy of a (regular) Riemannian foliation. This pseudogroup H (F) will then allow us to define a locally constant sheaf associated to F and hence to revisit the definition of singular Killing foliations from this viewpoint. In addition we will stress that blow-ups of singular Killing foliations are (regular) Killing foliations. Before we present the desired linearized holonomy pseudogroup, we are going to define a subpseudogroup that contains holonomy information near to the closure of a leaf. Consider the closure J := L of a leaf L ∈ F and let (U i , π i , γ ij ) be a Haefliger cocycle for the regular foliation F| J . For each U i , choose a point x i ∈ U i ⊂ J and a local reduction N i ⊂ M of F along a slice 36 for F| J containing x i . Furthermore, choose a distinguished tubular neighborhood Tub i ⊂ M of x i so that U i = J ∩ Tub i . As in the case of slices, we can suppose that there are submersions ρ i : Tub i N i whose fibers are dim(F| J )-dimensional and contained in the leaves of F. The compositions of the submersions ρ i with linearized flows with respect to J induce local diffeomorphisms with source and target contained in the disjoint union N := i N i and hence generate a pseudogroup of local diffeomorphisms H J acting on N . Consider F N i = F ∩ N i The local diffeomorphisms of H J project to local diffeomorphisms of the quotients N i /F N i and hence generate pseudogroups of local isometries on the disjoint unions of these metric spaces.
Lemma 10.4. With the notation established above, the following holds.
(i) (H J , i N i /F N i , g) is a complete metric pseudogroup with respect to the induced metric from g (ii) C (H J , i N i /F N i ) is a well defined locally constant sheaves of the infinitesimal transformations.
Proof outline. (i) Restricted to the regular part of the quotients N i /F N i , the pseudogroup H J is a subpseudogroup of the usual holonomy pseudogroup 6 . Therefore, since F restricted to the regular stratum is a regular Riemannian foliation, H J acts isometrically on the regular part of N i /F N i . By density of the regular points, we infer that H J acts isometrically everywhere. The completeness of the pseudogroup follows from the fact that the linearized flows that define the elements that generate H J are defined on open sets of J and hence their sources can be extended to the whole of N i . (ii) Consider the desingularization ρ : (M B , F B ) (M, F). By Proposition 9.3, local transverse isometries of F can be lifted to local transverse isometries of F B . Since each h ∈ H J is an extension of a holonomy of a leaf in F reg , we see that the lifting ρ * h is an element of the complete holonomy pseudogroup of the regular foliation F B . Hence ρ * H J is a complete pseudogroup and ρ * H J is a closed complete pseudogroup. We also have that ρ * H J = ρ * H J (see [8,Lemma 4.2]). Finally note that ρ * : H J ρ * H J is injective, because the pseudogroups are isometric. These facts together imply that C (H J , i N i /F N i ) is a well defined locally constant sheaf.
Remark 10.5. It is possible to check that the construction of the pseudogroup, and hence of the sheaf near J does not depend on the choice of the reduced spaces N i nor on the neighborhoods U i . Remark 10.6 (Geometric interpretation). From [7], [6] and the fact that F is infinitesimally closed we know that each element of the sheaf C (H J , i N i /F N i ) can be associated to a smooth transverse Killing vector field on a neighborhood of J that is tangent to F ⊂ F. Moreover this sheaf describes the transverse orbits of F on a neighborhood of J.
Once we have constructed a sheaf that describes the closure of F on a neighborhood of a fixed J = L, we are ready to extend it to a sheaf that describes the closure of F on M . We can cover M with a finite number of tubular neighborhoods of J γ = L γ and in each of these tubular neighborhoods we can consider the above construction. Therefore we have a natural disjoint union γ i∈Iγ N i that we can just write as j N j . Setting F N j as N j ∩ F we consider the disjoint union of metric spaces j N j /F N j . Now in order to define the pseudogroup H on j N j /F N j , consider two fixed leaf closures J i = L i and J i+1 = L i+1 and two reduced spaces N i and N i+1 associated to J i and J i+1 , respectively. Also assume that there exists a linearized flow ϕ i t (with respect to J i ) so that ϕ i+1,i := π i+1 •ϕ i t i is a map with source in N i and target in N i+1 (here π i+1 : Tub i+1 N i+1 is the projection whose fibers are contained in the leaves of F and such that dim π −1 i+1 (y) = dim F| J i+1 . In contrast with the previous construction of elements of H J , this map ϕ i+1,i does not need to be a bijective map. Nevertheless it can be projected and hence induces a local isometry (ϕ i+1,i ) * from N i /F N i to N i+1 /F N i+1 . Definition 10.7 (Linearized holonomy pseudogroup). The pseudogroup H generated by the local isometries of j N j /F N j described above is the linearized holonomy pseudogroup of F. Lemma 10.8 (Linearized holonomy pseudogroup). With the notation established above, the following holds: (i) The linearized holonomy pseudogroup (H , j N j /F N j , g) is a metric pseudogroup. (ii) Each h ∈ H with source in a reduced space N associated to J = L admits an extension h ∈ H J . In particular (H , j N j /F N j , g) is complete.
Proof outline. (i) The proof that the pseudogroup is a metric pseudogroup is similar to the one of Lemma 10.4.
Remark 10.9. The proof of item (ii) also allow us to check that the pseudogroup is well defined, that is it does not depend on the choice of J γ or the reduced spaces. On M reg the sheaf C clearly coincides with C reg , hence it coincides with the Molino sheaf C F on the whole M .
Corollary 10.11. The strong Molino conjecture holds for infinitesimally closed singular Riemannian foliations on compact manifolds. In particular, such a foliation is a singular Killing foliation if the manifold is simply connected.
In the next lemma we investigate how C F behaves under blow ups.
Lemma 10.12. Let F be an infinitesimally closed singular Riemannian foliation of a compact manifold M , and consider the k-th blow up ρ k+1 : (M k+1 , F k+1 ) (M k , F k ) of the sequence of blow ups of (M, F). Then C F k+1 is isomorphic to the inverse image ρ −1 k+1 C F k .
Proof outline. It is sufficient to show that C k+1 is the inverse image ρ −1 k+1 C k . Let U ⊂ M k+1 be a tubular neighborhood of a reduced space N where C k+1 coincides with the lift of C (H J ), for some leaf closure J k+1 . Suppose that U is small enough so that C k+1 is constant on U and there exists a similar tubular neighborhood V ⊂ M k containing ρ k+1 (U ) where C k is also constant and the lift of some C (H J k ). In this case we have ρ −1 k+1 C k (U ) = C k (V ), so we are done if we establish that each X ∈ C k+1 (U ) is the lift of some section in C k (V ), and that this correspondence is an isomorphism.
For this purpose we have to check that for s exp(sX) ∈ H k+1 we can find a dense set of {s i } so that exp(s i X) ∈ ρ * k+1 H k . First note that, if two F-flows ϕ 1 and ϕ 2 on M k+1 (i.e., flows whose integral lines are contained in F k+1 ) have homotopic integral lines contained in a leaf L k+1 ∈ F k+1 , then they induce the same holonomy (ϕ 1 ) * = (ϕ 2 ) * in H k+1 . The second fact is that, since F k+1 is infinitesimal closed, we can assume without loss of generality that the if integral line of a F-flow is contained in L k+1 ∩ ρ −1 k+1 (c), then the holonomy induced by this flow is the identity in H k+1 . These two facts together allow us to conclude that each h = exp(sX) ∈ H k+1 can be induced by a F-flow φ, with an integral line t φ t (x) transverse to the fibers of ρ k+1 , wherex ∈ ρ −1 k+1 (c). Therefore we can define α(t) := ρ k+1 (φ t (x)). Consider a linearized flow ϕ (with respect to Σ) with α as an integral line. Then its lift (ϕ) induces the holonomy h, in other words h ∈ ρ * k+1 H k . From Lemma 10.12 we infer the next result. Hence, desingularization provides another possible strategy for adapting results from the theory of regular Killing foliations to the singular setting.