Let $\mathcal F$ be a Lie foliation on a closed manifold $M$ with structural Lie group $G$. Its transverse Lie structure can be considered as a transverse action $Φ$ of $G$ on $(M,\mathcal F)$; i.e., an ``action'' which is defined up to leafwise homotopies. This $Φ$ induces an action $Φ^*$ of $G$ on the reduced leafwise cohomology $\bar H(\mathcal F)$. By using leafwise Hodge theory, the supertrace of $Φ^*$ can be defined as a distribution $L_{dis}(\mathcal F)$ on $G$ called the Lefschetz distribution of $\mathcal F$. A distributional version of the Gauss-Bonett theorem is proved, which describes $L_{dis}(\mathcal F)$ around the identity element. On any small enough open subset of $G$, $L_{dis}(\mathcal F)$ is described by a distributional version of the Lefschetz trace formula.
We study the transversal hard Lefschetz theorem on a transversely symplectic foliation. This article extends the results of transversally symplectic flows (H.K.~Pak, "Transversal harmonic theory for transversally symplectic flows", J. Aust. Math. Soc. 84 (2008), 233--245) to the general transversely symplectic foliation.
Let $F$ be a smooth foliation on a closed Riemannian manifold $M$, and let $Λ$ be a transverse invariant measure of $F$. Suppose that $Λ$ is absolutely continuous with respect to the Lebesgue measure on smooth transversals. Then a topological definition of the $Λ$-Lefschetz number of any leaf preserving diffeomorphism $(M,F)\to(M,F)$ is given. For this purpose, standard results about smooth approximation and transversality are extended to the case of foliation maps. It is asked whether this topological $Λ$-Lefschetz number is equal to the analytic $Λ$-Lefschetz number defined by Heitsch and La…
A foliation $(M,\mathcal{F})$ is said to be $2$--calibrated if it admits a closed 2-form $ω$ making each leaf symplectic. By using approximately holomorphic techniques, a sequence $W_k$ of $2$--calibrated submanifolds of codimension--$2$ can be found for $(M, \mathcal{F}, ω)$. Our main result says that the Lefschetz hyperplane theorem holds for the pairs $(F, F \cap W_k)$, with $F$ any leaf of $\mathcal{F}$. This is applied to draw important consequences on the transverse geometry of such foliations.
In this paper, we study the twisted basic Dolbeault cohomology and transverse hard Lefschetz theorem on a transverse Kahler foliation. And we give some properties for $Δ_κ$-harmonic forms and prove the Kodaira-Serre type duality for the twisted basic Dolbeault cohomology.
In this paper we try to generalize the Haefliger theorem on completly solvable Lie foliations. We prove that: every completely solvable Lie foliation on a compact manifold is the inverse image of a homogenus foliation. Every manifold in this paper is compact and our Lie group G is connexe and simply connexe.
A singular (or Hermann) foliation on a smooth manifold $M$ can be seen as a subsheaf of the sheaf $\mathfrak{X}$ of vector fields on $M$. We show that if this singular foliation admits a resolution (in the sense of sheaves) consisting of sections of a graded vector bundle of finite type, then one can lift the Lie bracket of vector fields to a Lie $\infty$-algebroid structure on this resolution, that we call a universal Lie $\infty$-algebroid associated to the foliation. The name is justified because it is isomorphic (up to homotopy) to any other Lie $\infty$-algebroid structure built on any ot…
We study left-invariant foliations $\mathcal{F}$ on Riemannian Lie groups $G$ generated by a subgroup $K$. We are interested in such foliations which are conformal and with minimal leaves of codimension two. We classify such foliations $\mathcal{F}$ when the subgroup $K$ is one of the important $\textbf{SU}(2)\times\textbf{SU}(2)$, $\textbf{SU}(2)\times\textbf{SL}_2(\mathbb{R})$, $\textbf{SU}(2)\times\textbf{SO}(2)$ or $\textbf{SL}_2(\mathbb{R})\times\textbf{SO}(2)$. By this we yield new multi-dimensional families of Lie groups $G$ carrying such foliations in each case. These foliations $\math…
We study left-invariant foliations ${\mathcal F}$ on semi-Riemannian Lie groups $G$ generated by a subgroup $K$. We are interested in such foliations which are conformal and with minimal leaves of codimension two. We classify such foliations ${\mathcal F}$ when the subgroup $K$ is one of the important $\text{SU}(2)$, $\text{SL}_{2}(\mathbb R)$, $\text{SU}(2)\times\text{SU}(2)$, $\text{SU}(2)\times\text{SL}_{2}(\mathbb R)$, $\text{SU}(2)\times\text{SO}(2)$, $\text{SL}_{2}(\mathbb R)\times\text{SU}(2)$. This way we construct new multi-dimensional families of Lie groups $G$ carrying such foliatio…
R. Zimmer proved that, on a compact manifold, a foliation with a dense leaf, a suitable leafwise Riemannian symmetric metric and a transverse Lie structure has arithmetic holonomy group. In this work we improve such result for totally geodesic foliations by showing that the manifold itself is arithmetic. This also gives a positive answer, for some special cases, to a conjecture of E. Ghys.
We associate a Lie $\infty$-algebroid to every resolution of a singular foliation, where we consider a singular foliation as a locally generated $\mathscr{O}$-submodule of vector fields on the underlying manifold closed under Lie bracket. Here $\mathscr{O}$ can be the ring of smooth, holomorphic, or real analytic functions. The choices entering the construction of this Lie $\infty$-algebroid, including the chosen underlying resolution, are unique up to homotopy and, moreover, every other Lie $\infty$-algebroid inducing the same foliation or any of its sub-foliations factorizes through it in an…
We consider all connected and simply connected 7-dimensional Lie groups whose Lie algebras have nilradical $\g_{5,2} = \s \{X_1, X_2, X_3, X_4, X_5 \colon [X_1, X_2] = X_4, [X_1, X_3] = X_5\}$ of Dixmier. First, we give a geometric description of the maximal-dimensional orbits in the coadjoint representation of all considered Lie groups. Next, we prove that, for each considered group, the family of the generic coadjoint orbits forms a measurable foliation in the sense of Connes. Finally, the topological classification of all these foliations is also provided.
We explicitly compute the diffeomorphism group of several types of linear foliations (with dense leaves) on the torus $T^n$, $n\geq 2$, namely codimension one foliations, flows, and the so-called non-quadratic foliations. We show in particular that non-quadratic foliations are rigid, in the sense that they do not admit transverse diffeomorphisms other than $\pm \id$ and translations. The computation is an application of a general formula that we prove for the diffeomorphism group of any Lie foliation with dense leaves on a compact manifold. Our results generalize those of P. Donato and P. Igle…
We describe a local model for any Singular Riemannian Foliation in a neighbourhood of a closed saturated submanifold of a regular stratum. Moreover we construct a Lie groupoid which controls the transverse geometry of the linear approximation of the Singular Riemannian Foliation around these submanifolds. We also discuss the closure of this Lie groupoid and its Lie algebroid.
In 86, Ranjan questioned whether a submersion $π: G\to B$ from a compact simple Lie group with bi-invariant metric is a coset foliation or not, provided the submersion is Riemannian with totally geodesic fibers. Here we answer this question affirmatively, even when the submersion is defined only in an open subset of $G$ (assuming suitable hypothesis).
This thesis treats two main topics: calibrated symplectic foliations, and local Lie groupoids. Calibrated symplectic foliations are one possible generalization of taut foliations of 3-manifolds to higher dimensions. Their study has been popular in recent years, and we collect several interesting results. We then show how de Rham's theory of currents, and Sullivan's theory of structure currents, can be applied in trying to understand the calibratability of symplectic foliations. Our study of local Lie groupoids begins with their definition and an exploration of some of their basic properties. N…
Every singular foliation has an associated topological groupoid, called holonomy groupoid (see arXiv:math/0612370). In this note we exhibit some functorial properties of this assignment: if a foliated manifold $(M,\mathcal{F}_M)$ is the quotient of a foliated manifold $(P,\mathcal{F}_P)$ along a surjective submersion with connected fibers, then the same is true for the corresponding holonomy groupoids. For quotients by a Lie group action, an analog statement holds under suitable assumptions, yielding a Lie 2-group action on the holonomy groupoid.
The purpose of this paper is to show that any extension of a minimal Lie foliation on a compact manifold is a transversaly Riemannian g\h- foliation with trivial normal bundle. This result permits to classify the extensions of a minimal Lie foliation on a compact manifold from the Lie subgroups of its Lie group.
We show that if the structure algebra of a Riemannian foliation F on a closed manifold M is nilpotent, then the integral of the Álvarez class of (M,F) along every closed path is the exponential of an algebraic number. By this result and the continuity of the Álvarez class under deformations shown in arXiv:1009.1098v2, we prove that the Álvarez class and the geometrically tautness of Riemannian foliations on a closed manifold M are invariant under deformation, if the fundamental group of M has polynomial growth.
Motivated by questions of deformations/moduli in foliation theory, we investigate the structure of some groups of diffeomorphisms preserving a foliation. We give an example of a $C^\infty$ foliation whose diffeomorphism group is not a Lie group in any reasonable sense. On the positive side, we prove that the automorphism group of a transversely holomorphic foliation or a riemannian foliation is a strong ILH Lie goup in the sense of Omori.
We compute an $L_\infty$-algebroid structure on a projective resolution of some classes of singular foliations on a vector space $V$ induced by the linear action of some Lie subalgebra of $\mathfrak {gl}(V)$. This $L_\infty$-algebroid provides invariants of the singular foliations, and also provides a constant-rank replacement of the singular foliation. We do this by first explicitly constructing projective resolutions of the singular foliations induced by the natural linear actions of endomorphisms of $V$ preserving a subspace $W\subset V$, the Lie algebra of traceless endomorphisms, and the …
The purpose of this short paper is to further develop the theory of transverse generalized complex structures. We focus on proving some equivalent conditions to the basic $dd^{\mathcal{J}}$ -lemma. We justify our approach by describing the transverse symplectic structure in this language and relating the basic $dd^{\mathcal{J}}$-lemma to the surjectivity of the Lefschetz map. We also present a non-trivial example of a foliation endowed with a transverse generalized complex structure. Transverse generalized complex structures do not rely heavily on the existence of a bundle-like metric, which m…
We combine classic stability results for foliations with recent results on deformations of Lie groupoids and Lie algebroids to provide a cohomological characterization for rigidity of compact foliations on compact manifolds.
Consider a complex analytic manifold $X$ and a coherent Lie subalgebra $\shi$ of the Lie algebra of complex vector fields on $X$. By using a natural $\shd_X$-module $\shm_\shi$ naturally associated to $\shi$ and the ring (in the derived sense) $\rhom[\shd_X](\shm_\shi,\shm_\shi)$, we associate integers which measure the irregularity of the foliation associated with $\shi$.
We consider four dimensional Lie groups with left-invariant Riemannian metrics. For such groups we classify left-invariant conformal foliations with minimal leaves of codimension two. These foliations produce local complex-valued harmonic morphisms.
We consider 5-dimensional Lie groups with left-invariant Riemannian metrics. For such groups we give a partial classification of left-invariant conformal foliations with minimal leaves of codimension 2. These foliations produce local complex-valued harmonic morphisms.
In this paper we study topological properties of an integrable case for Euler's equations on the Lie algebra $\textrm{so}(4)$, which can be regarded as an analogue of the classical Kovalevskaya case in rigid body dynamics. In particular, for all values of the parameters of the system under consideration the bifurcation diagrams of the momentum mapping are constructed, the types of critical points of rank $0$ are determined, the bifurcations of Liouville tori are described and the loop molecules for all singular points of the bifurcation diagram are computed. It follows from the obtained result…
We express Witten's deformation of Morse functions using deformation to the normal cone and $C^*$-modules. This allows us to obtain asymptotics of the `large eigenvalues'. Our methods extend to Morse functions along a foliation. We construct the Witten deformation using any generic function on an arbitrary foliation on a compact manifold and establish the compactness of its resolvent. When the foliation has a holonomy invariant transverse measure we show that our result implies Morse inequalities obtained by Connes and Fack in a slightly more general situation.
This doctoral thesis has two objectives. The first objective is to introduce a notion of equivalence for singular foliations that preserves their transverse geometry and is compatible with the notions of Morita equivalence of the holonomy groupoids and the transverse equivalence for regular foliations that appeared in the 1980's. The second one is to describe the structures behind quotients of singular foliations and to connect these results with their associated holonomy groupoids. It also wants to give an introduction to the notion of singular foliations as given by Androulidakis and Skandal…
The results of this manuscript is the collection of my articles that I published during my PhD thesis. We show that there is an equivalence of categories between Lie-Rinehart algebras over a commutative algebra $\mathcal O$ and homotopy equivalence classes of negatively graded acyclic Lie $\infty$-algebroids. Therefore, this result makes sense of the universal Lie $\infty$-algebroid of every singular foliation, without any additional assumption, and for Androulidakis-Zambon singular Lie algebroids. This extends to a purely algebraic setting the construction of the universal $Q$-manifold of a l…
We prove that the basic intersection cohomology $ {I H}^{^{*}}_{_{\bar{p}}}{(M/\mathcal{F})}, $ where $\mathcal{F}$ is the singular foliation determined by an isometric action of a Lie group $G$ on the compact manifold $M$, is finite dimensional.
Let $(G,g)$ be a 4-dimensional Riemannian Lie group with a 2-dimensional left-invariant, conformal foliation $\F$ with minimal leaves. Let $J$ be an almost Hermitian structure on $G$ adapted to the foliation $\F$. We classify such structures $J$ which are almost Kähler $(\A\K)$, integrable $(\I)$ or Kähler $(\K)$. Hereby we construct several new multi-dimensional examples in each class.
In this paper, we consider exponential, connected and simply connected Lie groups which are corresponding to Lie algebras of dimension 7 such that the nilradical of them is 5-dimensional nilpotent Lie algebra $\mathfrak{g}_{5,2}$ in Table 1. In particular, we give a description of the geometry of the generic orbits in the coadjoint representation of some considered Lie groups. We prove that, for each considered group, the family of the generic coadjoint orbits forms a measurable foliation in the sense of Connes. The topological classification of these foliations is also provided.
We prove some general results about the relation between the 1-cocycles of an arbitrary Lie algebroid $A$ over $M$ and the leaves of the Lie algebroid foliation on $M$ associated with $A$. Using these results, we show that a ${\cal E}^1(M)$-Dirac structure $L$ induces on every leaf $F$ of its characteristic foliation a ${\cal E}^1(F)$-Dirac structure $L_F$, which comes from a precontact structure or from a locally conformal presymplectic structure on $F$. In addition, we prove that a Dirac structure $\tilde{L}$ on $M\times \R$ can be obtained from $L$ and we discuss the relation between the le…
We prove that an isometric action of a Lie group on a Riemannian manifold admits a resolution preserving the transverse geometry if and only if the action is infinitesimally polar. We provide applications concerning topological simplicity of several classes of isometric actions, including polar and variationally complete ones. All results are proven in the more general case of singular Riemannian foliations.
The holonomy Lie $\infty$-groupoid of a singular foliation I
Department of Mathematics, University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校数学系）；Institut Élie Cartan de Lorraine, UMR 7502, Université de Lorraine（洛林埃利嘉当研究所）
In this paper we study the Lie groupoids which appear in foliation theory. A foliation groupoid is a Lie groupoid which integrates a foliation, or, equivalently, whose anchor map is injective. The first theorem shows that, for a Lie groupoid G, the following are equivalent: - G is a foliation groupoid, - G has discrete isotropy groups, - G is Morita equivalent to an etale groupoid. Moreover, we show that among the Lie groupoids integrating a given foliation, the holonomy and the monodromy groupoids are extreme examples. The second theorem shows that the cyclic homology of convolution algebras …
Let $(G,g)$ be a 4-dimensional Riemannian Lie group with a 2-dimensional left-invariant, conformal foliation $\mathcal{F}$ with minimal leaves. Let $J$ be an almost Hermitian structure on $G$ adapted to the foliation $\mathcal{F}$. The corresponding Lie algebra $\mathfrak{g}$ must then belong to one of 20 families $\mathfrak{g}_1,\dots,\mathfrak{g}_{20}$ according to S. Gudmundsson and M. Svensson. We classify such structures $J$ which are almost Kähler $(\mathcal{A}\mathcal{K})$, integrable $(\mathcal{I})$ or Kähler $(\mathcal{K})$. Hereby, we construct 16 multi-dimensional almost Kähler fami…
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…
Consider the Hamiltonian action of a torus on a transversely symplectic foliation that is also Riemannian. When the transverse hard Lefschetz property is satisfied, we establish a foliated version of the Kirwan injectivity theorem, and use it to study Hamiltonian torus actions on transversely Kähler foliations. Among other things, we prove a foliated analogue of the Carrell--Liberman theorem. As an application, this confirms a conjecture raised by Battaglia--Zaffran on the basic Hodge numbers of symplectic toric quasifolds. Our methods also allow us to present a symplectic approach to the calc…
