aa r X i v : . [ m a t h . DG ] F e b THE BASIC DE RHAM COMPLEX OF A SINGULAR FOLIATION
DAVID MIYAMOTO
Abstract.
A singular foliation F gives a partition of a manifold M into leaves whosedimension may vary. Associated to a singular foliation are two complexes, that of thediffeological differential forms on the leaf space M/ F , and that of the basic differentialforms on M . We prove the pullback by the quotient map provides an isomorphism of thesecomplexes in the following cases: when F is a regular foliation, when points in the leaves ofthe same dimension assemble into an embedded (more generally, diffeological) submanifoldof M , and, as a special case of the latter, when F is induced by a linearizable Lie groupoid. Introduction
A singular foliation F of a manifold M is a partition of M into connected, weakly-embedded submanifolds, called leaves, of perhaps varying dimension, satisfying the followingsmoothness condition: for every x ∈ M contained in a leaf L x , there exist locally definedvector fields X i about x such that • the X i are tangent to the leaves, and • the X ix span T x L x We then take the complex of basic differential forms to consist of those α ∈ Ω • ( M ) such that ι X α = 0 and L X α = 0 , for all X tangent to the leaves . The set of F -basic forms, denoted Ω • b ( M, F ), is a de Rham subcomplex of differential forms.We relate this complex to one on the quotient (or leaf) space M/ F . While rarely a smoothmanifold - for example, the leaf space of the irrational flow on the torus is not even Hausdorff- M/ F is naturally a diffeological space (introduced in Section 2). As a diffeological space, M/ F comes equipped with a de Rham complex of diffeological differential forms, Ω • ( M/ F ),and the quotient π : M → M/ F is diffeologically smooth. Its pullback induces a one-to-onemorphism from diffeological forms into basic forms. We seek singular foliations for whichthe following property holds Property (P) . The pullback π ∗ : Ω • ( M/ F ) → Ω • b ( M, F ) is into (hence, an isomorphism). Diffeology plays an important role. For example, we prove the foliation of the torus by anirrational flow has property (P), and as previously mentioned its leaf space is not a manifold.More generally, we prove
Date : February 22, 2021. A) Regular foliations have property (P) (Theorem 5.10).(B) F has property (P) the union of leaves of the same dimension is an embedded (moregenerally, diffeological) submanifold (Theorem 5.15).(C) F has property (P) if the induced singular foliation on M r { x | dim L x = 0 } hasproperty (P) (Theorem 5.17).We proved (A) independently, then found Hector, Marc´ıas-Virg´os, and Sanmart´ın-Carb´on[5] proved it earlier. We use groupoid techniques to approach this problem. For a groupoidversion of property (P), replace ( M, F ) with a Lie groupoid G ⇒ M ; replace M/ F with M/ G ; and take the basic forms to be α such that s ∗ α = t ∗ α . Then, we arrive at (A) byproving: • property (P) is Morita-invariant, • the Holonomy groupoid has property (P) if and only if F does, • an ´etale groupoid with countably generated pseudogroup (e.g. the ´etale version ofthe Holonomy groupoid) has property (P).Hector et al. deal directly with the action of Haefliger’s holonomy pseudogroup on a completetransversal. Our methods constitute a generalization of Hector et al.’s approach, and wemake explicit some correspondences left as routine in [5].Item (B) has not appeared in the literature, and extends a few existing results. In [10],Karshon and Watts proved that given a Lie group G acting on M whose identity component G ◦ acts properly, the corresponding action groupoid G ⋉ M ⇒ M has property (P). We canalternatively get this as a consequence of (B). Palais [14] showed that proper group actionsadmit slices, or equivalently, that their associated groupoids are linearizable, in the sense ofbeing locally isomorphic to a linearized version of the action. In Appendix A, PropositionA.3, we show that this implies the singular foliation of M by the G ◦ orbits satisfies thehypothesis in (B), hence has property (P). The fact G ◦ is connected implies G ◦ ⋉ M ⇒ M also has property (P) (Proposition 5.6), and a lemma from [10] shows this suffices to conclude G ⋉ M ⇒ M has property (P).In [21], Watts extended [10] to prove that proper Lie groupoids have property (P). As-suming source-connectedness, we give another argument. Crainic and Struchiner [1] showedthat proper Lie groupoids are linearizable. By Proposition A.3, linearizability implies thehypotheses of (B) hold, hence the foliation of M by the orbits of G has property (P). Byconnectedness, this is equivalent to G having property (P).A key feature of the above arguments is that linearizability of a (source-connected) Liegroupoid suffices to guarantee it has property (P). Since properness is sufficient but notnecessary for linearizability, our class of examples for which (P) holds is distinct from thosefound in [10] and [21]. However, even linearizability is not a necessary condition for (P). Ourfinal result (C) shows that the situation of the 0-dimensional leaves do not impact whether F has property (P). For example, every singular foliation of R has property (P), and manyof these are not linearizable. ur paper is structured as follows. Sections 2, 3, and 4 review diffeology, singular foliations,and Lie groupoids, respectively. In Section 5, we state and prove our main results. AppendixA is a review of linearizability, and contains the proof that linearizable Lie groupoids satisfythe hypotheses in (B). Appendix B reviews pseudogroups. Acknowledgements.
I am grateful to Yael Karshon for introducing this question to me,and providing constant and positive support throughout the writing process. I also thankJordan Watts for directing me to sources on Lie groupoids. This research is partially sup-ported by the Natural Sciences and Engineering Research Council of Canada.2.
Diffeology
We use diffeology to handle singular spaces, so we review of its basic concepts. Ourreference is Iglesias-Zemmour’s book [8].2.1.
Diffeology and Manifolds.Definition 2.1 (Diffeology) . Let X be a set. A parametrization into X is a map from anopen subset of a Cartesian space into X . A diffeology on X is a set D of parametrizations,whose members are called plots , such that • Constant maps are plots. • If a parametrization P : U → X is such that about each r ∈ U , there is an open V ⊆ U and a plot Q : V → X such that P = Q | V , then P is a plot. • If P : U → X is a plot and V is an open subset of a Cartesian space, then for anysmooth F : V → U , the pre-composition F ∗ P is a plot.A space equipped with a diffeology is a diffeological space .The set of locally constant parametrizations into X , and the set of all parametrizations into X , are both diffeologies, called the respectively discrete and coarse . Every other diffeologysits between these two. A classical smooth manifold (1) M carries a canonical diffeology D M ,consisting of the smooth maps (in the usual sense) from Cartesian spaces into M . Definition 2.2 (Smooth maps) . We say a map f : X → Y between diffeological spaces is diffeologically smooth if for every plot P of X , the pre-composition P ∗ f is a plot of Y . Denotethe smooth maps from X to Y by C ∞ ( X, Y ). If Y = R , we write C ∞ ( X ) := C ∞ ( X, R ).If ( X, D X ) is discrete or ( Y, D Y ) is coarse, all maps X → Y are smooth. A map betweenclassical manifolds is diffeologically smooth if and only if it is smooth in the usual sense. (1) a topological space locally homeomorphic to R n for some n , equipped with a compatible smooth atlas.We do not assume Hausdorff nor second-countable here. efinition 2.3 (D-topology) . The D-topology on a diffeological space X is the finest topol-ogy in which all plots are continuous. In other words, A ⊆ X is D-open if and only if P − ( A )is open for all plots P .Every diffeologically smooth map is continuous with respect to the D-topology. For aclassical manifold M , the D-topology associated to D M is the original topology on M . TheD-topology lets us define local objects in diffeology, in particular locally smooth maps, localdiffeomorphisms, and diffeological manifolds. Definition 2.4. • A partially defined function f : A ⊆ X → X ′ is locally smooth if A is D-open, andfor every for every plot P of X , the pre-composition P ∗ f : P − ( A ) → X ′ is a plot.Denote the set of locally smooth maps by C ∞ loc ( X, X ′ ). • If f : A ⊆ X → X ′ is locally smooth, injective, and f − : f ( A ) ⊆ X ′ → X is locallysmooth, we call f a local diffeomorphism . Denote the set of local diffeomorphisms byDiff loc ( X, X ′ ). • For fixed n , if about every x ∈ X there is a local diffeomorphism f : A ⊆ X → R n ,we call ( X, D ) a diffeological manifold of dimension n .The charts of a classically smooth n -manifold M are local diffeological diffeomorphismsinto R n . Therefore ( M, D M ) is a diffeological n -manifold. Conversely, the set of local diffe-ological diffeomorphisms from a diffeological n -manifold X into R n is a maximal atlas for asmooth structure σ on X that is compatible with the D-topology. Thus ( X, σ ) is a classical(perhaps not Hausdorff nor second-countable) n -manifold. This correspondence describes afull, faithful functor from the category of classical manifolds to that of diffeological spaces,whose image is the diffeological manifolds.Beyond classical manifolds, the following three spaces inherit a natural diffeology: subsetsof a diffeological space, quotients of a diffeological space, and function spaces of diffeologicallysmooth maps. Definition 2.5 (Subset diffeology) . • For a subset S of a diffeological space X , with inclusion ι : S ֒ → X , the subsetdiffeology on S consists of all parametrizations P : U → S such that ι ◦ P is a plotof X . The D-topology of the subset diffeology is not necessarily the subset topology. • A subset S is a diffeological submanifold of X if it is a diffeological manifold whenequipped with the subset diffeology.We take a moment to relate diffeological submanifolds to the more common notion ofweakly-embedded submanifolds: Definition 2.6.
A subset S of a classical smooth manifold M is a weakly-embedded subman-ifold if it is a diffeological submanifold of M , and the inclusion ι : S ֒ → M is an immersion. emark . If S is weakly-embedded, it admits a unique topology and smooth structure ofa classical smooth manifold. Therefore, when a subset is weakly-embedded, we may speakof it as a (sub)manifold without ambiguity.A natural question arises: is every diffeological submanifold of a classical smooth manifoldalso weakly-embedded? Perhaps surprisingly, the answer is no. Here is a counter-example,proposed by Y. Karshon and J. Watts. Take M = R , and set S := { ( x, y ) ∈ R | x = y } . Proposition 2.8.
The map f : R → S given by f ( t ) = ( t , t ) is a diffeological diffeomor-phism, where S carries the subset diffeology. Hence S is a diffeological submanifold. Observing that the inclusion is not an immersion at (0 ,
0) completes the example. Theproof of Proposition 2.8 invokes a 1982 result of H. Joris [9], which states that for everyparametrization Q : V → R , the composition f ◦ Q is smooth (as a map to R ) if and onlyif Q is smooth. Joris’ theorem also answers an open question in diffeology posed by Iglesias-Zemmour. Details will appear in an upcoming paper written jointly with Y. Karshon andJ. Watts. Definition 2.9.
Given a diffeological space X and a relation R with quotient map π : X → X/ R , the quotient diffeology on X/ R consists of those parametrizations P : U → X/ R suchthat about each r ∈ U , there is an open V ⊆ U and a plot Q : V → X of X such that P | V = π ◦ Q . In a diagram, Xr ∈ V U X/ R π ∃ Q P
The D-topology of the quotient diffeology always coincides with the quotient topology..
Definition 2.10 (Functional diffeology) . Let X and Y be diffeological spaces. The standardfunctional diffeology on C ∞ loc ( X, Y ) consists of those parametrizations P : U → C ∞ loc ( X, Y )satisfying: about each r ∈ U and x ∈ dom P ( r ), there are open neighbourhoods V ⊆ U ,and (D-open) A ⊆ dom P ( r ) such that • A ⊆ dom P ( r ) for all r ∈ V , and • the map V × A → Y given by ( r, x ) P ( r )( x ) is smooth.We equip C ∞ ( X, Y ) with the subset diffeology induced from C ∞ loc ( X, Y ). This is thecoarsest diffeology in which the evaluation map X × C ∞ ( X, Y ) → Y given by ( x, f ) f ( x )is smooth. We similarly equip Diff loc ( X, Y ) and Diff(
X, Y ) with their subset diffeologies. Inthe global case, the composition map is diffeologically smooth. For a classical manifold M ,we can prove the inversion map on Diff( M ) is also smooth. Proposition 2.11.
The map
Diff( M ) → Diff( M ) given by g g − is smooth. roof. Let P : U → Diff( M ) be a plot, meaning P : U → C ∞ ( M, M ) is a plot with imagein Diff( M ). It suffices to show r P ( r ) − is smooth as a map into C ∞ ( M, M ), so we prove(2.1) U × M → M, ( r, x ) P ( r ) − ( x )is smooth. Let P : U × M → U × M be the map ( r, x ) ( r, P ( r )( x )), and similarly define P − . Since P is a plot, P is smooth. Furthermore, it is a submersion: fixing ( r , x ) ∈ U × M and y := P ( r )( x ), the map M → U × M given by y ( r , P ( r ) − ( y )) is a smooth (since r is fixed) section of P through ( r , x ). Therefore by the inverse function theorem, locally P admits a smooth inverse. But this inverse is exactly P − , so this map, and hence (2.1), issmooth. (cid:3) Because composition and inversion are smooth, we call Diff( M ) a diffeological group . Anysubgroup of Diff( M ) with the subset diffeology is also a diffeological group. It is an openquestion whether Diff( X ) is a diffeological group for arbitrary diffeological spaces..2.2. Diffeological Forms.
Now we introduce diffeological differential forms, and state twouseful results for quotients.
Definition 2.12 (Diffeological forms) . A diffeological k-form α on X is an assignment toeach plot P : U → X a differential k -form α ( P ) ∈ Ω k ( U ) such that for every open subset V of a Cartesian space, and every smooth map F : V → U , we have α ( P ◦ F ) = F ∗ ( α ( P )) . Denote the set of diffeological k -forms by Ω k ( X ), and the set of diffeological forms by Ω • ( X ).As with usual differential forms, diffeologial forms pull back under smooth functions. Definition 2.13.
Let f : X → Y be smooth, and α ∈ Ω k ( Y ). The pullback f ∗ α ∈ Ω k ( X ) isdefined on the plots of X by ( f ∗ α )( P ) := α ( f ◦ P ).The set Ω k ( X ) is naturally a real vector space: given α, β ∈ Ω k ( X ) and λ ∈ R , define forevery plot P : U → X , ( λα + β )( P ) := λα ( P ) + β ( P ) . The space Ω • ( X ) also carries a differential d and wedge product ∧ , respectively defined by( dα )( P ) := dα ( P ) , ( α ∧ β )( P ) := α ( P ) ∧ β ( P ) . With respect to the grading Ω • ( X ) = L ∞ k =0 Ω k ( X ), the space (Ω • ( X ) , d, ∧ ) is a differen-tial commutative graded algebra. The pullback by a smooth function is a morphism ofcommutative differential graded algebras.Consider a classical manifold M . To each usual differential form α we may associatea diffeological form α by α ( P ) := P ∗ α . Conversely, to each diffeological form α , we canspecify a usual form α by declaring that in each chart ϕ of M , we have ϕ ∗ α := α ( ϕ ). Byidentifying α and α , we identify the usual de Rham complex on M with the diffeologicalone. Under this identification, the pullback by a smooth function viewed in the classical and iffeological senses agree. Therefore, we may freely switch between viewing classical formson M as diffeological ones, and vice versa.The next two results appear in [8] as Articles 6.38 and 6.39, respectively. Proposition 2.14.
Fix a diffeological space X with relation R , and equip X/ R with thequotient diffeology. The image of π ∗ : Ω k ( X/ R ) → Ω k ( X ) is those k -forms α on X suchthat for any two plots P, Q : U → X with π ◦ P = π ◦ Q , we have α ( P ) = α ( Q ) . Remark . Proposition 2.14 is equivalent to the following statement. Consider the dia-gram X × π X XX X/ R pr pr ππ Then α is in the image of π ∗ if and only if pr ∗ α = pr ∗ α . Equivalently, α is basic withrespect to the diffeological relation groupoid X × π X ⇒ X (c.f. Section 5).We omit the proof. Lemma 2.16.
The pullback π ∗ : Ω k ( X/ R ) → Ω k ( X ) is injective.Proof. The pullback π ∗ is a linear map, so it suffices to prove its kernel is trivial. Suppose β ∈ Ω k ( X/ R ) satisfies π ∗ β = 0. To show β = 0, we must show β ( P ) = 0 for any plot P : U → X/ R . We may do this locally. Let r ∈ U , and by definition of the pushforwarddiffeology, take a plot Q : V → X locally lifting P about r . Then β ( P ) | V = β ( P | V ) = β ( π ◦ Q ) = ( π ∗ β )( Q ) = 0 . So β ( P ) vanishes locally, hence on U . (cid:3) Singular Foliations
We take the view that a singular foliation of a manifold is a partition of the manifold intoconnected immersed submanifolds assembling together smoothly in some sense. To developthis sense, we use the notion of a smooth singular distribution on a manifold. This is aslightly different, but equivalent, approach to that taken in the introduction.Fix a classical manifold M , here always assumed to be Hausdorff and second-countable.We will drop the word “classical” for this section. Let V ( M ) denote the collection of vectorfields of M defined over open domains, and let X ( M ) denote the collection of globally definedvector fields. efinition 3.1. • A rough singular distribution ∆ over M is an assignment to each x ∈ M a linearsubspace ∆ x of the tangent space T x M . • A section of ∆ is any smooth vector field X ∈ V ( M ) such that X y ∈ ∆ y for every y ∈ dom X . Denote the set of sections of ∆ by Γ loc (∆). • If for every x ∈ M and v ∈ ∆ x , there is a section of ∆ through v , i.e. some X ∈ Γ loc (∆)such that X x = v , then we say ∆ is smooth . Remark . An equivalent condition for smoothness is: about each x ∈ M , there are vectorfields X i ∈ V ( M ) defined near x such that span( X ix ) = ∆ x and for all i and all y ∈ dom( X i ),we have X iy ∈ ∆ y . This is the analogue of the definition of smooth singular foliation presentedin the introduction.An relevant question is: given a smooth singular distribution, about each x ∈ M , do thereexist vector fields X i ∈ V ( M ) defined on U ∋ X such that span( X iy ) = ∆ y for all y ∈ U ?This was resolved in the affirmative independently by Sussmann [20], and by Drager, Lee,Park, and Richardson (DLPR) [2]. We will use this fact later, so state it here. Theorem 3.3 (Sussman-DLPR) . Given a smooth singular distribution ∆ , there always exists X i ∈ X ( M ) such that span( X ix ) = ∆ x for all x ∈ M . Moreover, the X i can be chosen to becomplete. For a smooth singular distribution, consider the dimension function M → N , x dim ∆ x . This lets us classify points as regular or singular: points x ∈ M for which there is a neigh-bourhood U on which x dim ∆ x is constant are the regular points , and all others are singular . We also introduce the subsets M ∗ k of M , for ∗ ∈ { = , ≥ , >, <, ≤ , = } , defined by M ∗ k := { x ∈ M | dim ∆ x ∗ k } . The dimension map is lower semi-continuous, so the sets M ≥ k are open. In particular, onecan show the set of regular points is open and dense in M . Definition 3.4. A singular foliation of M is a partition F of M such that its members,called leaves , are weakly-embedded and connected, and the associated singular distribution(∆ F ) x := T x L is smooth. Here L is the leaf of F containing x . Remark . For weakly-embedded submanifolds, see Definition 2.6. Since the weakly-embedded submanifolds admit unique smooth structures, a singular foliation does indeeddepend only on the partition of M , and not on any choice of structure on its leaves. Also,because we assume M is Hausdorff and second-countable here, it follows that the leaves areHausdorff and second-countable, too.We call a leaf regular or singular according to whether its points are regular or singularwith respect to ∆ F . The union of regular leaves is open and dense in M . tefan in [18] and Sussmann in [19] independently made groundbreaking progress in thestudy of singular foliations. Here we briefly review their results. Stefan originally workedwith what he called “arrows” on a manifold M . While his terminology was not widelyadopted, these are natural objects in diffeology. Definition 3.6. An arrow on a manifold M is a plot a : U ⊆ R → Diff loc ( M ) such that a (0) = id and if x ∈ dom a ( t ), then x ∈ dom a ( s ) for every 0 ≤ s ≤ t . Note a ( t ) might bethe empty map.If X ∈ V ( M ) has flow Ψ, the map t Ψ( t, · ) is an arrow. However, in general we do notrequire arrows to satisfy a group law. Take a collection of arrows A on M , and set • Ψ A to be the pseudogroup (see Appendix B) generated by S a ∈A a ( R ). • (∆ A ) x := (cid:26) ddt (cid:12)(cid:12)(cid:12) t = t a ( t, y ) | a ∈ A , a ( t , y ) = x (cid:27) . • (∆ A ) x := { dϕ ϕ − ( x ) ( v ) | ϕ ∈ Ψ A , v ∈ (∆ A ) ϕ − ( x ) } .One can verify that both ∆ A and ∆ A are smooth singular distributions. The followingappears as Theorem 1 in [18]. Theorem 3.7 (Stefan) . Let A be a collection of arrows on a manifold M . The orbits of Ψ A are leaves of a singular foliation of M . For each leaf, T x L = (∆ A ) x . Now we discuss Sussmann’s contribution. This next result can also be derived from Ste-fan’s theorem after a bit of work, and is therefore called the Stefan-Sussmann theorem. Wefirst require the notion of an integrable smooth singular distribution.
Definition 3.8.
Fix a smooth singular distribution ∆. • An integral submanifold of ∆ through x ∈ M is an immersed submanifold L contain-ing x such that for every y ∈ L , we have T y L = ∆ y . • An integral submanifold L is maximal if it is connected, and every other connectedintegral submanifold intersecting L is an open submanifold of L . If a maximal integralsubmanifold exists about x , it is unique. • We say ∆ is integrable if it admits an integral submanifold at each point in M .It is immediate from the definition of a singular foliation that the associated smoothsingular distribution ∆ F is integrable. The Stefan-Sussmann theorem essentially states thatall integrable singular distributions are ∆ F for some singular foliation F . Theorem 3.9 (Stefan-Sussmann) . Fix a smooth singular distribution ∆ . The following areequivalent.(a) ∆ is integrable.(b) There is a maximal integral submanifold about each point of M . c) Let D be a “spanning set” of locally-defined vector fields, meaning ∆ x = span { X x | X ∈ D} , and take A to be those arrows of the form Ψ( t, · ) , where Ψ is the flow ofsome X ∈ D . Then ∆ = ∆ A .In the case of (a), (b), or (c), the orbits of Ψ A are exactly the maximal integral submanifoldsof ∆ . By Stefan’s theorem, these are weakly-embedded, and therefore form a singular foliation F of M such that ∆ = ∆ F . Sussmann called the condition ∆ = ∆ A “ D -invariance” of ∆. Stefan called it “homogene-ity” of A . Historically, it was only Stefan who noted the maximal integral submanifoldsof ∆ are not just immersed, but weakly-embedded. It was only Sussmann who noted themaximality property of these orbits.As a corollary to the Stefan-Sussmann theorem, we can show that every singular foliationis uniquely determined by its associated integrable singular distribution. This generalizesthe Frobenius theorem for regular foliations, in the following sense: for a regular foliation F ,the Frobenius implies that correspondence F 7→ ∆ F between foliations of M and integrableregular distributions is a bijection. Corollary 3.10.
The map { singular foliations } → { integrable singular distributions }F 7→ ∆ F is a bijection.Proof. Smoothness of ∆ F comes from the definition of singular foliation, and the leaves F witness integrability. Now for integrable ∆, define F ∆ to be the partition of M intoits maximal integral submanifolds. This is a well-defined singular foliation by the Stefan-Sussmann theorem. It is now straightforward to check ∆
7→ F ∆ and F 7→ ∆ F are inverses. (cid:3) Let us finally give some examples of singular foliations.
Example . Fix any smooth non-negative bounded function f : R → R , and let C := f − (0). Consider the smooth singular distribution ∆ on R spanned by D := { f ∂∂x } . Thepoints of C and the connected components of R − C constitute maximal integral submanifoldsof ∆, hence form a singular foliation of R . The set M =0 is just C , and M =1 is R − C . Asexpected, M =1 is open, and the regular points are int( C ) ∪ M =1 , which is dense in R .In fact, every closed subset of R may be realized as C above. Therefore the collection ofsingular leaves of a singular foliation may be quite wild. For instance, there is a singularfoliation of R whose singular points form a Cantor set.By a similar argument, the partition of M into maximal integal curves of any completevector field X ∈ X ( M ) is a singular folaition of M . xample . Let G ⇒ M be a Lie groupoid over M , with associated Lie algebroid ( ρ, g )(see the next section for Lie groupoids and algebroids). Take F to be the partition of M into the connected components of the orbits of G , each equipped with their canonical smoothstructure. (2) Then the distribution associated to F is exactly ρ ( g ), and one can show this issmooth. Therefore by definition, F is a singular foliation.Since the connected component of the identity G ◦ ⇒ M has the same Lie algebroid as G , Corollary 3.10 implies the singular foliation induced by G ◦ is equal to F . But the leavesof the former are exactly the orbits of G ◦ , since these are already connected. Therefore theconnected components of the G orbits are the G ◦ orbits.We can apply this discussion to obtain the same statements for a Lie group acting smoothlyon M , by considering the action groupoid. If we consider the R action on M induced by theflow of a complete vector field, we recover the first example. Example . Given an arbitrary Lie algebroid ( ρ, A ), one can show the distribution ρ ( A )is integrable using the Stefan-Sussmann theorem. But historically, Hermann in 1962 [6] gavea sufficient condition for integability that predates Stefan and Sussmann’s work, is easier tocheck, and is satisfied by ρ ( A ). This example subsumes the previous two. It is unknownwhether every integrable distribution is of the form ρ ( A ) for some Lie algebroid A .4. Lie Algebroids and Groupoids
Bibundles and Morita Equivalence.
Our reference for this review is [11]. A
Liegroupoid is a small category G ⇒ G with invertible arrows, such that the base G is aHausdorff, second-countable smooth manifold, the arrow space G is a smooth manifold (notnecessarily Hausdorff or second-countable), all structure maps are smooth, and furthermorethe source map is a smooth submersion with Hausdorff fibers.We denote the source-fiber at x by P x := s − ( x ) and the isotropy group at x by G x := s − ( x ) ∩ t − ( x ). Because s is a submersion, P x is an embedded submanifold of G . It can alsobe shown that G x is a Lie group. The orbit of G through x ∈ G is O := t ( P x ). This is anequivalence class of the relation x ∼ y if there is an arrow x y . We can equip O with acanonical smooth structure such that t : P x → O is a principal G x -bundle (see Theorem 5.4in [13]), and O is immersed in G . Denote the orbit space by M/ G . Example . Our first example of a Lie groupoid is the action groupoid associated to a Liegroup G acting smoothly on a manifold M . The base is M , and the arrow space is G × M .The source and target maps are s ( g, x ) = x, t ( g, x ) = g · x. The unit map is u ( x ) = ( e, x ), and the inversion is ( g, x ) − = ( g − , x ). The multiplicationis given by ( h, y )( g, x ) = ( hg, x ), and is defined whenever g · x = y . We denote the actiongroupoid by G ⋉ M . Its orbits are exactly the orbits of G . (2) that is, the structure induced by viewing t : s − ( x ) → O x as a principal G x -bundle ust as a Lie group can act on a manifold, we also define a (right) Lie groupoid action ona manifold. Definition 4.2. A right action of a Lie groupoid H on a manifold P consists of an anchormap a R : P → H and a multiplication P × a R t H → P fitting into the diagram P × a R t H P H H µ pr pr a R s t Moreover ( p · h ) · h ′ = p · ( hh ′ ) whenever this makes sense, and p · | a ( p ) = p for all p ∈ P .We can view P × a R t H ⇒ P as a Lie groupoid itself, denoted P ⋊ H , with source map µ and target pr . The multiplication is ( p, h )( p ′ , h ′ ) = ( p, hh ′ ). Note ( a R , pr ) is a morphismof Lie groupoids P ⋊ H → H . Definition 4.3. A principal right H bundle is a surjective submersion π : P → B anda right H action on P such that π ( p · h ) = π ( p ) whenever this makes sense, and H actsfreely and transitively on the fibers of π . Precisely, the map P × a R t H → P × B P given by( p, h ) ( p, p · h ) is a diffeomorphism. Remark . We can view the bundle P × B P → P as a Lie groupoid P × B P ⇒ P , wherethe first and second projections constitute the source and target maps. When we have aprincipal right H bundle, we get a morphism of Lie groupoids P ⋊ H → P × B P that is theidentity on the base and the diffeomorphism ( p, h ) ( p, p · h ) on arrows. Example . Every Lie groupoid H ⇒ H acts on its arrow space H by right multiplicationalong the anchor map s : H → H . In other words, P = H , a R = s , and µ ( h, h ′ ) = hh ′ .Equipping H with this action, the surjective submersion t : H → H becomes a principalright H bundle.We can analogously define the left action of a Lie groupoid G on P (denoted G ⋉ P ), anda principle left G bundle. Left and right actions allow us to define a bibundle. Definition 4.6. A bibundle between two groupoids G and H is a triple ( P, a L , a R ), where P is a manifold, a L : P → G and a R : P → H , and • G acts on P from the left, with anchor a L , and H acts on P from the right, withanchor a R . • a L : P → G is a principal right H -bundle. • a R is G -invariant. • the actions of G and H commute.We also write P : G → H . n isomorphism of bibundles P, Q : G → H is a diffeomorphism α : P → Q that isequivariant with respect to G − H action, i.e. α ( g · p · h ) = g · α ( p ) · h , whenever this makessense. It is also possible to define a non-associative composition of bibundles. Using Liegroupoids as the objects, bibundles as the 1-arrows, and isomorphisms of bibundles as the2-arrows, we get a weak 2-category, sometimes denoted Bi . See [11] for details. Definition 4.7.
Two Lie groupoids G and H are Morita equivalent if there is a bibundle P : G → H such that a R : P → H is a principal left G -bundle. In this case we can invert P to get P − : H → G . Remark . An alternative way to define Morita equivalence is through the use of refine-ments of Lie groupoids. We call a functor φ : G → H a refinement if • it is essentially surjective (ES), meaning the map t ◦ pr : H × s φ G → H is asurjective submersion, and • it is fully faithful (FF), meaning the square G H G × G H × H φ ( s,t ) ( s,t )( φ × φ ) is a pullback.Then G and H are Morita equivalent if and only if there is a groupoid K and refinements K → G and
K → H . The invertible bibundle induced by a refinement φ : G → H is thepullback along φ of the the principal right H -bundle t : H → H from Example 4.5; herethe pullback of a right principal H -bundle π : P → B along a map F : N → B is the rightprincipal H -bundle pr : F ∗ P → N , where • F ∗ P = P × B N is the pullback of the bundle π : P → B by F : N → B , • the anchor map for the right H action is a R ◦ pr , • the action is µ ( n, p, h ) = ( n, p · h ).4.2. The holonomy groupoid.
For this section we use [13]. We call a Lie groupoid G ⇒ G source-connected if the source-fibers are all connected. As with Lie groups, we can considerthe source-connected identity component G ◦ ⇒ G of a Lie groupoid. The arrows in G ◦ arethose arrows g ∈ G such that g and 1 s ( g ) belong to the same component of s − ( s ( g )). Thisis an open source-connected subgroupoid of G (see [12]).A Lie groupoid is ´etale if the source map is ´etale. To each ´etale Lie groupoid G ⇒ G , wehave an associated pseudogroup Ψ( G ) on G (see Appendix B for more on pseudogroups).This is defined in terms of bisections of G . A bisection of G is a local section σ of s such that t ◦ σ is a diffeomorphism. Then we setΨ( G ) := { t ◦ σ | σ is a bisection of G} . ow fix a regular folaition F on M . Its holonomy groupoid is a Lie groupoid Hol ⇒ M ,with arrows x y the holonomy classes of leafwise paths, and multiplication given byconcatenation of paths. Hol is source-connected.Fix a complete transversal ι : S ֒ → M , i.e. an embedded submanifold of M meetingevery leaf transversally. We can pull back Hol along ι (equivalently, restrict it to S ) to getHol S ⇒ S , a Lie groupoid whose arrows are those arrows in Hol with endpoints in S . ThisLie groupoid is ´etale and Morita equivalent (3) to Hol. We can then consider the associatedpseudogroup Ψ(Hol S ) on S . This is exactly the classical holonomy pseudogroup associatedto F , for which the next result is fundamental. Theorem 4.9.
For a regular foliation F with complete transversal S , the holonomy pseu-dogroup Ψ(Hol S ) is countably generated. Basic de Rham complexes
Definition 5.1.
Fix a singular foliation ( M, F ) with associated singular distribution ∆. Adifferential form α ∈ Ω • ( M ) is F - basic if for every section X ∈ Γ loc (∆),(5.1) ι X α = 0 , and L X α = 0 . A form satisfying the first condition is horizontal , and the second is invariant . Denote theset of basic forms by Ω • b ( M, F ). Remark . We could potentially define, for any smooth singular distribution ∆, a ∆-basicform to be one which satisfies (5.1) for all X ∈ Γ loc (∆). While there are no technical problemswith this definition, if ∆ is not integrable, then by Corollary 3.10 there is no associatedsingular foliation F , and hence no related leaf-space. Since our goal is to investigate how thebasic forms capture transverse structures, it therefore does not make sense for us to definebasic in this generality.Because L X α = ι X ( dα ) − d ( ι X α ), any horizontal form is invariant if and only if ι X dα = 0.In other words, α ∈ Ω • b ( M, F ) if and only if ι X α = 0 and ι X dα = 0 for all X ∈ Γ loc (∆). Butthe interior derivative ι X at x depends only on X x . We therefore conclude that to prove α is F -basic, it suffices to check (5.1) against any set of vector fields spanning ∆. In particular,given a Lie algebroid ( g , ρ ), a form is g -basic (i.e. (5.1) holds for all X ∈ ρ (Γ( g ))) if and onlyif it is basic with respect to the induced singular foliation.The differential and exterior product of basic forms remain basic, so Ω • b ( M, F ) is a sub-complex of Ω • ( M ). In the presence of a Lie groupoid, there is another complex of “basic”forms. Definition 5.3.
Fix a Lie groupoid G ⇒ M . A differential form α ∈ Ω • ( M ) is G -basic if s ∗ α = t ∗ α . Denote the de Rham complex of these forms by Ω • b ( M, G ). (3) the inclusion is a refinement, hence induces a Morita equivalence. See Remark 4.8. emark . For a Lie group G acting on M , there is also a notion of a G -basic form. Thisis a form α such that g ∗ α = α for all g ∈ G , and ι X α for all X tangent to the orbits of G .However, this is not a fundamentally different notion than that from Lie groupoids: a formis G -basic if and only if it is G ⋉ M -basic (Lemma 3.3 in [21]).We relate the complexes of G -basic and F G -basic forms in Proposition 5.6.5.1. Basic forms and regular foliations.
In this section we prove statement (A) in theIntroduction, namely that for a regular foliation F , the pullback by the quotient π : M → M/ F is an isomorphism π ∗ : Ω • ( M/ F ) → Ω • b ( M, F ). Another proof can be found in [5], butwe arrived at this result independently.To contextualize the following propositions, we will outline the proof here. The pullbackis one-to-one for purely diffeological reasons (Lemma 2.16). First, we show π ∗ has imagecontained in F -basic forms, so that the question is well-posed (Proposition 5.5). Then,we show that the F -basic forms are exactly the Hol-basic forms, because Hol is a source-connected groupoid whose orbits are the leaves of F (Proposition 5.6). This reduces thequestion to whether π ∗ : Ω • ( M/ F ) → Ω • b ( M, Hol) is an isomorphism. Now take a completetransversal S to F , to obtain the ´etale holonomy groupoid Hol S ⇒ S , Morita equivalent toHol ⇒ M . Because of this equivalence, the pullback by π is an isomorphism if and only ifthe pullback by π S : S → S/ Hol S is an isomorphism (Proposition 5.7, Corollary 5.8). ButHol S has finitely generated pseudogroup, and we can show this implies π ∗ S is an isomorphism(Lemma 5.9). Proposition 5.5. (a) For a singular foliation, every pullback of a form on M/ F is F -basic.(b) For a Lie groupoid, every pullback of a form on M/ G is G -basic.Proof. (a) Suppose β ∈ Ω k ( M/ F ) and set α := π ∗ β . For any X ∈ Γ loc (∆), we must show ι X α = 0 and L X α = 0. By Proposition 2.14, for any plots P, Q : U → M such that π ◦ P = π ◦ Q , we have P ∗ α = Q ∗ α . We can in fact replace the domain of P, Q withany manifold, and thus set P := Φ : D → M to be the flow of X, ( t, p ) Φ t ( p ) = Φ ( p ) ( t ) Q := pr : D → M, ( t, p ) p. Since Φ t ( p ) and p always share a leaf, π ◦ Φ = π ◦ pr , and therefore Φ ∗ α = pr ∗ α .For ( t, p ) ∈ D , we identify T ( t,p ) D with R ⊕ T p M . Under this identification, for every k vectors v , . . . , v k ∈ T p M , at t = 0 we have Φ ∗ α ) (0 ,p ) (1 ⊕ v , v , . . . , v k ) = α p ( X p + v , v , . . . , v k )(pr ∗ α ) (0 ,p ) (1 ⊕ v , v , . . . , v k ) = α p ( v , v , . . . , v k ) . This implies ι X p α p = 0. We also have for each t , at ~v = ( v , . . . , v k ), α p ( ~v ) = (pr ∗ α ) ( p,t ) ( ~v ) = (Φ ∗ α ) ( t,p ) ( ~v ) = ((Φ t ) ∗ α ) p ( ~v ) , hence 0 = ddt (cid:12)(cid:12)(cid:12) t =0 α p ( ~v ) = ddt (cid:12)(cid:12)(cid:12) t =0 ((Φ t ) ∗ α ) p ( ~v ) = ( L X α ) p ( ~v )Therefore ι X α and L X α vanish.(b) This argument is Corollary 3.6 in [21]. Suppose β ∈ Ω k ( M/ G ) and set α := π ∗ β .Then by Remark 2.15, α is basic with respect to the relation groupoid M × π M ⇒ M . So then pr ∗ α − pr ∗ α = 0. Pulling this back by ( s, t ) : G → M × π M yields0 = ( s, t ) ∗ (pr ∗ α − pr ∗ α ), and thus s ∗ α = t ∗ α . (cid:3) For this next proposition, we adapt the proof from [7], which deals only with regularfoliations.
Proposition 5.6.
Let G ⇒ M be a Lie groupoid with associated singular foliation F . Every G -basic form is F -basic, and every F -basic form is G ◦ -basic.Proof. Let ( g , ρ ) be the Lie algebroid of G , where g = (ker ds ) | M and ρ = dt . To to prove aform is F -basic, it suffices to only test against vector fields in ρ (Γ( g )). For a section σ of g ,denote the corresponding right-invariant vector field on G by ˜ σ . For an arrow g : x y , wehave ds (˜ σ g ) = ds ( dR g (˜ σ y )) by definition of ˜ σ = d ( s ◦ R g )(˜ σ y )= 0 because s ◦ R g is constant . Similarly, dt (˜ σ g ) = dt (˜ σ y ). Therefore˜ σ ∼ s σ ∼ t ρ ( σ ) , which implies that for any form α on M ,(5.2) L ˜ σ s ∗ α = 0 , and L ˜ σ t ∗ α = t ∗ L ρ ( σ ) α. Now we take the splitting T G| M = g ⊕ T M , which allows us to write( s ∗ α ) x ( ξ + w , . . . , ξ k + w k ) = α x ( w , . . . , w k )( t ∗ α ) x ( ξ + w , . . . , ξ k + w k ) = α x ( ρ ( ξ ) + w , . . . , ρ ( ξ k ) + w x ) . (5.3)We will now prove the proposition. Suppose α is G -basic. Then s ∗ α = t ∗ α , and so setting v i = 0 for i > w = 0 inequation (5.3), we get α x (0 , w , . . . , w k ) = ( ι ρ ( ξ ) α ) x ( w , . . . , w k ) . The left side is always 0, so we get ι ρ ( σ ) α = 0. For invariance, by equation (5.2),0 = L ˜ σ s ∗ α = L ˜ σ t ∗ α = t ∗ L ρ ( σ ) α, and since t is a submersion, we get L ρ ( σ ) α = 0. Therefore α is F -basic. • Now suppose α is F -basic. The fact ι ρ ( σ ) α = 0 implies α x ( ρ ( ξ ) + w , . . . , ρ ( ξ k ) + w k ) = α x ( w , . . . , w k ) , so by equation (5.3) we get that s ∗ α = t ∗ α at points 1 x ∈ M . Furthermore theassumption L ρ ( σ ) α = 0 combined with equation (5.2) gives L ˜ σ s ∗ a = 0 = t ∗ L ρ ( σ ) α = L ˜ σ t ∗ α. Therefore s ∗ α and t ∗ α are invariant under the flows of all the ˜ σ . Now notice thatthese vector fields span ker ds , which is an involutive subbundle of T G that foliates G by the connected components of the source-fibers. In particular, we can connect anyarrow in the component 1 x to 1 x by travelling along the flows of the ˜ σ . Therefore s ∗ α = t ∗ α on the union of connected components of the identity arrows. This isexactly the arrow space of G ◦ ⇒ M , thus α is G ◦ -basic. (cid:3) The following proposition and its proof are from [21], Proposition 3.9, but can also befound as Lemma 5.3.8 in [7].
Proposition 5.7.
Let G and H be Morita equivalent Lie groupoids, witnessed by an invertiblebibundle P : G → H . There is an isomorphism P ∗ : Ω • b ( M, H ) → Ω • b ( M, G ) defined uniquelyby the condition that a ∗ R α = a ∗ L P ∗ α (where a R and a L are the anchor maps for the actions).Proof. The bibundle P : G → H gives the following commutative diagram.
G × s a L P P × a R t HG P H G H µ L pr pr pr µ R pr ts a R a L s t Let α ∈ Ω • b ( M, H ). The pullback a ∗ R α is P ⋊ H -basic, since by commutativity and H -basic, µ ∗ R a ∗ R α = pr ∗ s ∗ α = pr ∗ t ∗ α = pr ∗ a ∗ R α. y Remark 4.4, the Lie groupoid P ⋊ H is isomorphic to P × G P , so then a ∗ R α is also P × G P -basic. By Remark 2.15, this is equivalent to a ∗ R α = a ∗ L β , for some β ∈ Ω • ( G ).Note β is unique because a L is a surjective submersion. We say P ∗ α := β . We claim β is G -basic. First, observe that by commutativity, a ∗ R α = a ∗ L β , and a ∗ R α being G ⋉ P -basic:pr ∗ s ∗ β = pr ∗ a ∗ L β = pr ∗ a ∗ R α = µ ∗ L a ∗ R α = µ ∗ L a ∗ L β = pr ∗ t ∗ β. Then s ∗ β = t ∗ β because a L , hence pr , is a surjective submersion.It is evident that P ∗ is well-defined, and a homomorphism of complexes. To see it is anisomorphism, observe that its inverse is ( P − ) ∗ . (cid:3) Corollary 5.8.
The pullback π ∗G surjects onto G -basic forms if and only if π ∗H is onto H -basicforms.Proof. The map Ψ : H / H → G / G defined by π H ( y ) π G ( a L ( a − R ( y ))) is a well-defineddiffeological diffeomorphism ([21], Theorem 3.8). Now assume π ∗G surjects, and take α ∈ Ω • b ( M, H ). Set β := P ∗ α . By assumption there is some β ∈ Ω • ( G / G ) with π ∗G β = β . Then a ∗ L β = a ∗ L π ∗G β = a ∗ R ( π ∗H Ψ ∗ β ) . But a ∗ L β is also a ∗ R α , and since a R is a surjective submersion, we get α = π ∗H Ψ ∗ β . In otherwords, π ∗H also surjects. For the converse direction, work with ( P − ) ∗ . (cid:3) Now, we have a final lemma.
Lemma 5.9.
Let G ⇒ M be an ´etale Lie groupoid with countably generated associatedpseudogroup Ψ( G ) . Then the pullback by π : M → M/ G is onto G -basic forms.Proof. See Appendix B for the relevant facts about pseudogroups. Let α ∈ Ω • b ( M, G ). Take P, Q : U → M such that π ◦ P = π ◦ Q . By Proposition 2.14, it suffices to show P ∗ α = Q ∗ α .First, note that α is Ψ( G )-invariant, since for any f = t ◦ σ ∈ Ψ( G ), we have f ∗ α = σ ∗ t ∗ α = σ ∗ s ∗ α = id ∗ α = α. Say { f i } ∞ i =1 generates Ψ( G ). For each tuple I := ( i , . . . , i N ), define f I := f i ◦ · · · ◦ f i N , andset C I := { r ∈ U | f I ( P ( r )) = Q ( r ) } . Each C I is closed in U , and we claim U ⊆ S I C I (hence equality holds). Indeed, for any r ∈ U , we have π ( P ( r )) = π ( Q ( r )), so there is anarrow P ( r ) Q ( r ). Taking its image under Eff (see the discussion preceding DefinitionB.3), this gives some f ∈ Ψ( G ) such that f ( P ( r )) = Q ( r ). Using our generating family forΨ( G ), we can write f = f I locally near r for some I , hence r ∈ C I .By the Baire category theorem, S I int( C I ) is open and dense in U . But on each int( C I ),we have f I ◦ P = Q , so by Ψ( G )-invariance of α , P ∗ α = P ∗ f ∗ α = ( f I ◦ P ) ∗ α = Q ∗ α. As this holds on the open dense subset S I int( C I ), by continuity P ∗ α = Q ∗ α on all of U , asrequired. (cid:3) e may now give the formal statement and proof of our result (A). Theorem 5.10.
Suppose ( M, F ) is a regular foliation. Equip M and M/ F with their man-ifold and quotient diffeology, respectively. The quotient map π : M → M/ F is diffeologicallysmooth, and its pullback restricts to an isomorphism from diffeological forms on M/ F to F -basic forms on M . In other words, π ∗ : Ω • ( M ) → Ω • b ( M, F ) is an isomorphism (c.f. [5] ).Proof. The pullback π ∗ is injective by Lemma 2.16, and maps into basic forms by Proposition5.5 (a). It remains is to show π ∗ is surjective.The foliation F is induced by its holonomy groupoid Hol ⇒ M . As Hol is source-connected, by Proposition 5.6 the F -basic and Hol-basic forms coincide. Since M/ Hol = M/ F , it therefore suffices to show that π ∗ is onto Hol-basic forms.Fix a complete transversal S to F . The restriction Hol S ⇒ S of Hol to S is Moritaequivalent to Hol (see Section 4.2). Therefore by Corollary 5.8, to show π ∗ surjects onto Hol-basic forms, we may instead show the pullback by π S : S → S/ Hol S is onto Hol S -basic forms.The groupoid Hol S ⇒ S is ´etale, and its associated pseudogroup is countably generated byTheorem 4.9. We apply Lemma 5.9 to complete the proof. (cid:3) Corollary 5.11. If G ⇒ M is any source-connected Lie groupoid Morita equivalent to an´etale groupoid, (4) the pullback by π : M → M/ G is an isomorphism onto G -basic forms.Proof. Because G is Morita equivalent to an ´etale groupoid, its associated isotropy groupsare discrete (Proposition 5.20 in [13]). Therefore the singular foliation F associated to G is regular, and Theorem 5.10 every F -basic form is a pullback from the quotient. But the F -basic forms are exactly the G -basic forms by source-connectedness and Proposition 5.6.So the pullback must be onto G -basic forms as well. (cid:3) Singular foliations decomposed by dimension.
We now prove statement (B) fromthe Introduction. We begin by establishing some terminology.
Definition 5.12.
A singular foliation ( M, F ) is decomposed by dimension if the sets M = k ,consisting of points in leaves of dimension k , are diffeological submanifolds of M , perhapswith components of varying dimension.Take an arbitrary singular foliation ( M, F ) with associated distribution ∆. For ∗ ∈ { = , ≥ , >, <, ≤ , = } , set F ∗ k := { L ∈ F | dim L ∗ k } . Lemma 5.13. If M ∗ k is a diffeological submanifold of M , then ( M ∗ k , F ∗ k ) is a singularfoliation. (4) such groupoids are often called foliation groupoids . roof. By the Stefan-Sussmann Theorem 3.9, we may take a collection of arrows A suchthat the orbits of Ψ A are the leaves of F . Let A ′ consist of the arrows in A restricted to M ∗ k . This is a collection of arrows, and so by Stefan’s Theorem 3.7, the orbits of Ψ A ′ forma singular foliation of M ∗ k . But these orbits are exactly the elements of F ∗ k , hence F ∗ k isindeed a singular foliation. (cid:3) Lemma 5.14.
Suppose M ∗ k is a diffeological submanifold of M , and α ∈ Ω • b ( M, F ) . Then α ′ := α | M ∗ k is F ∗ k -basic.Proof. Take A and A ′ as in the previous lemma. Then α is F -invariant, hence Ψ A -invariant.This means α ′ is Ψ A ′ -invariant, hence F ∗ k -invariant. As for horizontal, we must take aslightly pedantic approach. Denote the inclusions L M ∗ k M. ι ′ ι ′′ ι Both ι ′ and ι ′′ are smooth immersions, since L is weakly-embedded, but ι is merely smooth.Suppose v ∈ T x L . We want to show ι ι ′∗ v α ′ = 0. Compute ι ι ′∗ v α ′ = α ′ ( ι ′∗ v, · ) = α ( ι ∗ ι ′∗ v, ι ∗ · ) = α ( ι ′′∗ v, ι ∗ · ) . But the right side is 0, because α is F -horizontal. (cid:3) Theorem 5.15.
Suppose the singular foliation ( M, F ) is decomposed by dimension. Equip M and M/ F with the manifold and quotient diffeology, respectively. The quotient map π : M → M/ F is diffeologically smooth, and pulling back by the quotient is an isomorphismfrom diffeological forms on M/ F to F -basic forms on M . In other words, π ∗ : Ω • ( M/ F ) → Ω • b ( M, F ) is an isomorphism.Proof. By Proposition 5.5, π ∗ maps into F -basic forms, and by Lemma 2.16 π ∗ is injective.It remains to show π ∗ is surjective. Let k max denote the highest dimension of the leaves of F . Equip M ∗ k with the singular foliation F ∗ k . Fix α ∈ Ω • b ( M, F ). Consider the statement S ( k ) α | M ≥ k is the pullback of a form on the quotient M ≥ k / F ≥ k . If S ( k max ) holds, and if S ( k + 1) = ⇒ S ( k ), then S (0) holds, which is what we want to prove.Now, S ( k max ) is equivalent to: there is a form β on M = k max / F = k max such that π ∗ β = α | M = k max .But ( M = k max , F = k max ) is a regular foliation, and α | M = k max is F = k max -basic by Lemma 5.14, so S ( k max ) holds by Theorem 5.10.Now assume S ( k +1). We will use Proposition 2.14 to conclude S ( k ). Let P, Q : U → M ≥ k be plots such that π ◦ P = π ◦ Q . Set A := P − ( M ≥ k +1 ) (= Q − ( M ≥ k +1 )) , which is open in UB := P − ( M = k ) (= Q − ( M = k )) . Then U = A ⊔ B , and so U = A ∪ int( B ). We will show α ( P ) = α ( Q ) first on A , and thenon int( B ). By continuity, this yields α ( P ) = α ( Q ) on A ∪ int( B ) = U . For A : The plots P and Q restrict to maps P ′ , Q ′ : A → M ≥ k +1 , which are smoothmaps between the U -open set A and the M -open set M ≥ k +1 . We have π ◦ P ′ = π ◦ Q ′ ,and we are assuming S ( k + 1). Therefore by Proposition 2.14, α | M ≥ k +1 ( P ′ ) = α | M ≥ k +1 ( Q ′ ) , which implies α ( P ) | A = α ( Q ) | A . • For int( B ): We may assume int( B ) is non-empty. Fix r ∈ int( B ). By the Stefan-Sussmann Theorem (Remark 2.7), we can take M -open sets V about P ( r ) and V ′ about Q ( r ), and a diffeomorphism ξ : V → V ′ respecting the leaves, such that ξ ( P ( r )) = Q ( r ). Let U ′ be an open connected neighbourhood of r such that U ′ ⊆ int( B ) ∩ P − ( V ). Define P ′ : U ′ → M = k , r ′ ξ ( P ( r ′ )) Q ′ : U ′ → M = k , r ′ Q ( r ′ ) . These are continuous. As U ′ is connected, its images under P ′ and Q ′ are connected,and in particular lie in a connected component of M = k . Because P ′ ( r ) = Q ′ ( r ), infact both P ′ and Q ′ map into a single connected component M ◦ = k of M = k . Denotethe restrictions of P ′ and Q ′ to maps U ′ → M ◦ = k again by P ′ and Q ′ . We may view P ′ and Q ′ as smooth maps from the U -open set U ′ to the manifold M ◦ = k . Observethat π ( P ′ ( r ′ )) = π ( ξ ( P ( r ′ ))) = π ( P ( r ′ )) = π ( Q ′ ( r ′ )) , so π ◦ P ′ = π ◦ Q ′ . Also, α | M ◦ = k is basic with respect to F = k by Lemma 5.14. Thereforeby Theorem 5.10 α | M ◦ = k is the pullback of some form on M ◦ = k / F = k . By Proposition2.14, we get α | M ◦ = k ( P ′ ) = α | M ◦ = k ( Q ′ ) , which implies α ( ξ ◦ P ) | U ′ = α ( Q ) | U ′ . Because α is F -basic, α ( ξ ◦ P ) = α ( P ). As r was arbitrary, we can conclude that α ( P ) | int( B ) = α ( Q ) | int( B ) .Therefore, for any two plots P, Q of M such that π ◦ P = π ◦ Q , we have proved α ( P ) = α ( Q ). By Proposition 2.14, α is the pullback of some diffeological form on M/ F . (cid:3) Corollary 5.16. If G ⇒ M is a source-connected linearizable Lie groupoid (see AppendixA), then π ∗ : Ω • ( M/ G ) → Ω • b ( M, G ) is an isomorphism.Proof. Let F G be the singular folaition consisting of the orbits of G . By Proposition 5.6, G linearizable implies F G is decomposed by dimension. Hence by Theorem 5.15, π is onto F G -basic forms. By source-connectedness, and Proposition 5.6, Ω • b ( M, F G ) = Ω • b ( M, G ), whichcompletes the proof. (cid:3) Here we discuss how this result fits into existing literature. Crainic and Struchiner [1]proved every proper Lie groupoid is linearizable. In this case, the fact π ∗ is an isomorphismwas also proved by Watts [21]. Watts relied on properness to ensure compactness of theisotropy groups G x , and then applied his and Karshon’s [10] earlier result that π ∗ is anisomorphism whenever G is the action groupoid of a Lie group action on M with properly cting identity component. This earlier result also relied on compactness of the G x for aproper Lie group action.To finish, we show that π ∗ is an isomorphism for a much broader class of singular foliationsthan those decomposed by dimension. Theorem 5.17.
Suppose the singular foliation ( M, F ) is such that the pullback by π : M > → M > / F > is an isomorphism π ∗ : Ω • ( M > ) → Ω • b ( M > , F > ) . Then the pullback by thequotient π : M → M/ F is an isomorphism π ∗ : Ω • ( M ) → Ω • b ( M, F ) .Proof. This proof uses the same ideas as in the previous theorem. Letting α ∈ Ω • b ( M, F ),all we need to show is that α comes from the quotient. We use Proposition 2.14. Let P, Q : U → M be plots such that π ◦ P = π ◦ Q . Set A := P − ( M > ) (= Q − ( M > )) , which is open in UB := P − ( M =0 ) (= Q − ( M =0 )) . As before, to show α ( P ) = α ( Q ), it suffices to show equality on A and on int( B ). For A ,this is a direct result of the assumption and the fact M > is open, so that ( M > , F > ) isa singular foliation of a manifold. For r ∈ int( B ), note that π ◦ P ( r ) = π ◦ Q ( r ) reducesto P ( r ) = Q ( r ), because the 0-leaves are just points. Therefore P = Q on int( B ), and α ( P ) = α ( Q ) on int( B ) follows immediately. (cid:3) Some consequences of the previous theorem are that π ∗ is an isomorphism when • ∆ F is spanned by a single vector field. For instance, every singular foliation of R . • the singular leaves of ( M, F ) are all points (this implies M = k is open for k ≥ , R ), and more generallyany singular foliation induced by any Poisson structure on a 2 or 3-dimensionalmanifold.Both examples can come from Lie groupoids which are not linearizable, and perhaps someare not induced by any Lie algebroid at all. Appendix A. Linearization of Lie groupoids and decomposition by dimension
Here we review the linearization of a Lie groupoid about an orbit. Our sources are [1]and [4]. Refer to Section 4 for the relevant facts about Lie groupoids. Fix a Lie groupoid G ⇒ M , and orbit O through x ∈ M . We form the restricted groupoid G O ⇒ O , whosearrows are those arrows in G which begin (and end) in O , and all the structure maps areinduced from G ⇒ M . This is a Lie groupoid. et ν ( O ) denote the normal bundle over O , and similarly for ν ( G O ). We can form thegroupoid ν ( G O ) ⇒ ν ( O ) using the short exact sequence of groupoids:1 T G O T G ν ( G O ) 10 T O T M ν ( O ) 0 Definition A.1.
A Lie groupoid is linearizable at an orbit O (through x ) if there is anopen neighbourhood U of O ⊆ M and an open neighbourhood V of O ⊆ ν ( O ) (viewing O as the image of the zero-section), and an isomorphism of the Lie groupoids G| U ⇒ U and ν ( G O ) | V ⇒ V , which is the identity on G O ⇒ O .We next present two alternative descriptions of the linear model. First, G O acts on ν ( O )from the left with anchor π by. µ L ( g, [ v ]) = g · [ v ] := [ dt g (˜ v )] where ˜ v ∈ T g G satisfies ds g (˜ v ) = v. The action is well-defined, and yields the action groupoid G O ⋉ ν ( O ). One can check thisis isomorphic to ν ( G O ) by [˜ v g ] ( g, [ ds g ˜ v ]). Our second description is point-wise. Considerthe two principal G x -bundles: t : P x → O P x × P x → G O , ( g, h ) gh − . The action µ L provides a left action of G x on ν x O . Then, we can form the two associatedbundles, P x × G x ν x O , and ( P x × P x ) × G x ν x O . Both bundles above are isomorphic to ν ( O ) and G O ⋉ ν ( O ), respectively: the isomorphismsare [ k, w ] k · w , and [ k, k ′ , w ] ( k ( k ′ ) − , k · w ). Using these identifications to transport thegroupoid structure of ν ( G O ), we obtain the Lie groupoid ( P x × P x ) × G x ν x O ⇒ P x × G x ν x O ,which is a pointwise version of the linear model. Example
A.2 . Consider a Lie group G acting on a manifold M , with orbit O through x .Take G to be the action groupoid G ⋉ M . Its pointwise linear model is given by the left G -action on G × G x ν x O . If the action is locally proper at x (see Definition 2.3.2 in [3]),the Tube Theorem (for example, Theorem 2.4.1 in [3]) gives a G -equivariant diffeomorphismfrom G × G x D , where D is a G x -invariant open neighbourhood of 0 in ν x O , to some saturatedneighbourhood U of O in M . Therefore if the action is locally proper at x , it is linearizablein the sense of Definition A.1 at O , and furthermore we can take U and V saturated. Palais[14] showed that a Lie group action with compact isotropy is linearizable at O if and only ifif it locally proper at x . rainic and Struchiner in [1] proved that a Lie groupoid G ⇒ M is linearizable at theorbit O through x if it is proper at x . Note that a Lie groupoid may be proper at each x ∈ M , hence linearizable at each orbit, without being globally proper.Of interest to us is the fact every linearizable Lie groupoid induces a singular foliation thatis decomposed by dimension. In the context of proper groupoids, this fact is also proved byPosthuma, Tang, and Wang in [17]. The authors rely the linearization theorem for properLie groupoids given by [1], as well as a description of the linear model provided in [15].Because linearizable Lie groupoids are a key example for us, and we prefer to avoid assumingproperness, we give a self-contained - but not fundamentally dissimilar - reproduction of thisfact. Proposition A.3.
A linearizable Lie groupoid induced singular folaition decomposed by di-mension.Proof.
Fix x ∈ M , with orbit O . By linearizability, get open U and V about O , and anisomorphism of the Lie groupoids G| U ⇒ U and ( G O ⋉ ν ( O )) | V ⇒ V . Since the dimensionof the orbit about y ∈ U is entirely determined by dim G y , we relate G y to G x .Say y corresponds to v . Viewing ν ( O ) as P x × G x ν x O , we may find unique [ k, w ] in theassociated bundle such that k · w = v . We propose the following diffeomorphism:( G O ⋉ ν ( O )) ⊇ iso( v ) → k stab G x ( w ) k − ⊆ G O , ( g, v ) g. • Well-defined: the only arrows from v in the action groupoid are of the form ( g, v ),because the source map is just the projection of the second coordinate. Also, bydefinition of the action, g : π ( v ) π ( g · v ) = π ( v ) k : x π ( k · w ) = π ( v ) , hence k − gk ∈ G x . Moreover k − gk · w = k − g · v = k − · v = w. So finally, we can write g = k ( k − gk ) k − ∈ k stab G x ( w ) k − . • Smooth: this is the restriction of a projection, and both domain and codomain areembedded submanifolds. • Inverse: the inverse map g ( g, v ) is well-defined because g = kγk − for some γ ∈ stab G x ( w ), hence g · v = kγk − · v = kγ · w = k · w = v. It is smooth since it is the restriction of an inclusion.Therefore, G y ∼ = iso( v ) ∼ = k stab G x ( w ) k − . In particular, dim G y = dim stab G x ( w ), anddim G y = dim G x if and only if stab G x ( w ) is an open submanifold of G x about the identity.In this case, necessarily stab G x ( w ) ⊇ G ◦ x , where G ◦ x is the identity component of G x . In ther words, the set of w fixed by G ◦ x corresponds exactly to those y such that dim G y =dim G x . Denote by ( ν x O ) G ◦ x the vector subspace of fixed points of G ◦ x . In this notation,we conclude the diffeomorphism U → V from the linearization descends to a bijection M = k ∩ U → V ∩ P x × G x ( ν x O ) G ◦ x . The codomain is an embedded submanifold of V , and thuswe take these bijections as charts for an atlas of M = k . These make M = k into an embeddedsubmanifold of M . The components of M = k may have different dimensions. (cid:3) Example
A.4 . A consequence of the proof above is that if a Lie groupoid is linearizable at x ,there is a neighbourhood of the orbit through x such that all isotropy groups are conjugateto a subgroup of G x . We can use this to provide a linear action which is not linearizable.Consider the representation of SL ( C ) on binary forms of degree 3 (i.e. polynomials of theform p ( x, y ) = P a i x − i y i ), given by M · p ( x, y ) := p ( M ( x, y ) T ). This is the unique irreduciblerepresentation of SL ( C ) in 4 (complex) dimensions. The isotropy of x y is trivial, yet inany neighbourhood of x y there is a form with isotropy of order at least 3; this is nontrivial,see [16] page 162. Therefore the action groupoid is not linearizable around the orbit of x y . Appendix B. Pseudogroups
A pseudogroup is a group of locally-defined diffeomorphisms, and these arise naturallyin the context of singular foliations. For this discussion, fix a smooth manifold M . If f : U → U ′ and g : V → V ′ are two smooth functions on open subsets of M , denote by g ◦ f the restricted composition g ◦ f : f − ( V ) → g ( U ′ ); note we restrict the codomain. As a specialcase, for a subset W of dom f , denote by f | W the restriction f | W : W → f ( W ). We allow forcompositions to be the empty map. Finally, given a set of maps A , let A − := { f | f − ∈ A } . Definition B.1. A transition on M is a diffeomorphism f : U → U ′ of open subsets of M .A pseudogroup P is a collection of transitions such that(i) id | U is in P for every open subset U .(ii) If f and f ′ are in P , then so are f ′ ◦ f and f − .(iii) If f : U → U ′ is a transition and { U i } i ∈ I is a cover of U , and we have f | U i in P foreach i ∈ I , then f is in P .The intersection of an arbitrary collection of pseudogroups is itself a pseudogroup. Inparticular, given an arbitrary set of transitions A , the pseudogroup generated by A is definedas P A := \ A ⊆ P ′ P ′ , where P ′ runs over all pseudogroups containing A . It is the minimal pseudogroup containing A . If P is a pseudogroup and A is a set of transitions with P A = P , we say P is generated by A . If A can be chosen countable (finite), we say P is countably (finitely) generated. A pseu-dogroup generated by A consists exactly of those transitions that are locally compositionsof elements of A ∪ A − , in the following sense. emma B.2. Suppose P is a pseudogroup generated by a set of transitions A . Then f : U → U ′ is in P if and only if about each p ∈ U , there is an open neighbourhood V in U andtransitions f , . . . , f n ∈ A ∪ A − such that f | V = ( f ◦ · · · ◦ f n ) | V . The proof is straightforward.B.1.
The germ groupoid.
Here assume M is Hausdorff and second countable. Associatedto each pseudogroup P on M is the germ groupoid Γ P , a Lie groupoid. In the sequel wedenote Γ P by Γ for simplicity. Its base manifold is Γ = M . Its set of arrows Γ consists ofall germs of transitions in P , that isΓ := { germ x f | f ∈ P, x ∈ dom f } . The source of germ x f is x , and the target is f ( x ), so that germ x f : x f ( x ). Themultiplication is given by the composition of germs: if f, g ∈ P and f ( x ) ∈ dom g , thengerm f ( x ) g · germ x f := germ x ( f ◦ g ). The unit x x is the germ of the identity, and theinverse of germ x f is germ f ( x ) f − . With these structures, it is not hard to see Γ is a groupoid.To see it is Lie, we require a smooth structure on Γ such that the source map s is asubmersion with Hausdorff fibers, and the other structure maps are smooth. Given ( f : U → U ′ ) ∈ P , consider the map U → Γ , x germ x f, and denote its restriction to its image by ˜ f . Let Ψ := { ˜ f | f ∈ P } . We claim Ψ is an atlas(with values in M ) for Γ making Γ ⇒ M an ´etale Lie groupoid.Recall in Section 4.2 we associated to each Lie groupoid G ⇒ M a pseudogroup Ψ( G )on M , whose elements were t ◦ σ for all local bisections σ . We have Ψ(Γ( P )) = P . As forΓ(Ψ( G )), there is a natural morphism Eff : G → Γ(Ψ( G )) which is the identity on the base,and maps the arrow g to Eff( g ) := germ s ( g ) ( t ◦ σ ), where σ is a section of s through g (whosegerm is unique because s is ´etale). This functor (on arrows) is onto, but is not injective ingeneral. Definition B.3.
We call Γ(Ψ( G )) the effect of G , and denote it by Eff( G ). An ´etale groupoid G is effective if the functor Eff : G → Γ(Ψ( G )) is injective, and hence an isomorphism of Liegroupoids. References [1] Marius Crainic and Ivan Struchiner,
On the linearization theorem for proper Lie groupoids , Ann. Sci.´Ec. Norm. Sup´er. (4) (2013), no. 5, 723–746.[2] Lance D. Drager, Jeffrey M. Lee, Efton Park, and Ken Richardson, Smooth distributions are finitelygenerated , Ann. Global Anal. Geom. (2012), no. 3, 357–369.[3] Johannes J. Duistermaat and Johan A. C. Kolk, Lie Groups , Universitext, Springer, Berlin, 2000.
4] Rui L. Fernandes,
Normal forms and Lie groupoid theory , Geometric Methods in Physics (Bia lowie˙za,Poland, 2014), Trends in Mathematics, Birkh¨auser, Basel, 2015, pp. 49–66.[5] Gilbert Hector, Enrique Marc´ıas-Virg´os, and Esperanza Sanmart´ın-Carb´on,
De Rham cohomology ofdiffeological spaces and foliations , Indag. Math. (N.S.) (2011), no. 3–4, 212–220.[6] Robert Hermann, The differential geometry of foliations II , J. Math. Mech (1962), no. 2, 303–315.[7] Benjamin Hoffman and Reyer Sjamaar, Stacky Hamiltonian actions and symplectic reduction , with anappendix by Chenchang Zhu (2019), preprint, to appear. arXiv.org:1808.01003v3.[8] Patrick Iglesias-Zemmour,
Diffeology , Mathematical Surveys and Monographs, vol. 185, American Math-ematical Society, Providence, 2013.[9] Henri Joris,
Une C ∞ -application non-immersive qui poss`ede la propri´et´e universelle des immersions ,Arch. Math. (1982), no. 3, 269–277 (French).[10] Yael Karshon and Jordan Watts, Basic forms and orbit spaces: a diffeological approach , SIGMA Sym-metry Integrability Geom. Methods Appl. , 19 pp.[11] Eugene Lerman, Orbifolds as stacks? , Enseign. Math. (2) (2010), no. 3–4, 315–363.[12] Kirill C. H. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids , London MathematicalSociety Lecture Note Series, vol. 213, Cambridge University Press, Cambridge, 2005.[13] Ieke Moerdijk and Janez Mrc˘un,
Introduction to foliations and Lie groupoids , Cambridge Studies inAdvanced Mathematics, vol. 91, Cambridge University Press, Cambridge, 2003.[14] Richard S. Palais,
On the existence of slices for actions of non-compact Lie groups , Ann. of Math. (1961), no. 2, 295–323.[15] Marcus J. Pflaum, Hessel Posthuma, and Xiang Tang, Geometry of orbit spaces of proper Lie groupoids ,J. Reine Angew. Math. (2014), 49–84.[16] Vladimir L. Popov and Ernest B. Vinberg,
Algebraic Geometry IV . 2:
Invariant Theory (Aleksei P.Parshin and Igor R. Shafarevich, eds.), Encyclopaedia of Mathematical Sciences, vol. 55, Springer-Verlag, Berlin, 1994.[17] Hessel Posthuma, Xiang Tang, and Wang Kirsten,
Resolutions of proper Riemannian Lie groupoids , Int.Math. Res. Not. IMRN (2021), 1249–1287.[18] Peter Stefan, Accessible sets, orbits, and foliations with singularities , Proc. London Math. Soc. (3) (1974), 699–713.[19] H´ector J. Sussmann, Orbits of families of vector fields and integrability of distributions , Trans. Amer.Math. Soc. (1973), 171–188.[20] ,
Smooth distributions are globally finitely spanned , Analysis and Design of Nonlinear ControlSystems: In honor of Alberto Isidori (Alessandro Astolfi and Marconi Lorenzo, eds.), Springer-Verlag,Berlin, pp. 3–8.[21] Jordan Watts,
The orbit space and basic forms of a proper Lie groupoid , 2015, preprint, to appear.arXiv:1309.3001v4., 2015, preprint, to appear.arXiv:1309.3001v4.