Actions of totally disconnected groups and equivariant singular homology

We study equivariant singular homology in the case of actions of totally disconnected locally compact groups on topological spaces. Theorem A says that if G is a totally disconnected locally compact group and X is a G-space, then any short exact sequence of covariant coefficient systems for G induces a long exact sequence of corresponding equivariant singular homology groups of the G-space X. In particular we consider the case where G is a totally disconnected compact group, i.e., a profinite group, and G acts freely on X. Of special interest is the case where G is a p-adic group, p a prime. The conjecture that no p-adic group, p a prime, can act effectively on a connected topological manifold, is namely known to be equivalent to the famous Hilbert-Smith conjecture. The Hilbert-Smith conjecture is the statement that, if a locally compact group G acts effectively on a connected topological manifold M, then G is a Lie group.     

Compact Kähler quotients of algebraic varieties and Geometric Invariant Theory

Given an action of a complex reductive Lie group G on a normal variety X, we show that every analytically Zariski-open subset of X admitting an analytic Hilbert quotient with projective quotient space is given as the set of semistable points with respect to some G-linearised Weil divisor on X. Applying this result to Hamiltonian actions on algebraic varieties, we prove that semistability with respect to a momentum map is equivalent to GIT-semistability in the sense of Mumford and Hausen. It follows that the number of compact momentum map quotients of a given algebraic Hamiltonian G-variety is finite. As further corollary we derive a projectivity criterion for varieties with compact Kähler quotient.     

A complete stability theorem for foliations with singularities

We study codimension one smooth foliations with singularities on closed manifolds. We assume that the singularities are nondegenerate (of Bott-Morse type) in the sense of Scárdua and Seade (2009) [9] and prove a version of Thurston-Reeb stability theorem in terms of a component of the singular set: If all singularities are of center type and the foliation exhibits a compact leaf with trivial Cohomology group of degree one or a codimension ?3 component of the singular set with trivial Cohomology group of degree one then the foliation is compact and stable.     

Actions of Totally Disconnected Groups and Equivariant Singular Cohomology

In this work, we study the special properties of the equivariant singular cohomology of a G-space X, where G is a totally disconnected, locally compact group. We prove that any short exact sequence of coefficient systems for G, over a ring R, gives a long exact sequence of the associated equivariant singular cohomology modules. We establish the relationship between the ordinary singular cohomology modules and the equivariant singular cohomology modules with the natural contravariant coefficient system. Moreover, under some conditions, we give an isomorphism of the equivariant singular cohomology modules of the G-space X onto the ordinary singular cohomology modules of the orbit space X/G.     

Finiteness of the basic intersection cohomology of a Killing foliation

We prove that the basic intersection cohomology ${{\mathbb H}^{*}_{\overline{p}}({M / \mathcal{F}})}$ , 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.     

Reduced Delzant spaces and a convexity theorem

The convexity theorem of Atiyah and Guillemin-Sternberg says that any connected compact manifold with Hamiltonian torus action has a moment map whose image is the convex hull of the image of the fixed point set. Sjamaar-Lerman proved that the Marsden-Weinstein reduction of a connected Hamitonian G-manifold is a stratified symplectic space. Suppose 1→A→G→T→1 is an exact sequence of compact Lie groups and T is a torus. Then the reduction of a Hamiltonian G-manifold with respect to A yields a Hamiltonian T-space. We show that if the A-moment map is proper, then the convexity theorem holds for such a Hamiltonian T-space, even when it is singular. We also prove that if, furthermore, the T-space has dimension and T acts effectively, then the moment polytope is sufficient to essentially distinguish their homeomorphism type, though not their diffeomorphism types. This generalizes a theorem of Delzant in the smooth case. This paper is a concise version of a companion paper [B. Lian. B. Song, A convexity theorem and reduced Delzant spaces, math.DG/0509429].     

Laplacians and Spectrum for Singular Foliations

The author surveys Connes＇ results on the longitudinal Laplace operator along a （regular） foliation and its spectrum, and discusses their generalization to any singular foliation on a compact manifold. Namely, it is proved that the Laplacian of a singular foliation is an essentially self-adjoint operator （unbounded） and has the same spectrum in every （faithful） representation, in particular, in L2 of the manifold and L2 of a leaf. The author also discusses briefly the relation of the Baum-Connes assembly map with the calculation of the spectrum.     

Quantisation commutes with reduction at discrete series representations of semisimple groups

Using the analytic assembly map that appears in the Baum-Connes conjecture in noncommutative geometry, we generalise the Guillemin-Sternberg conjecture that ‘quantisation commutes with reduction’ to (discrete series representations of) semisimple groups G with maximal compact subgroups K acting cocompactly on symplectic manifolds. We prove this generalised statement in cases where the image of the momentum map in question lies in the set of strongly elliptic elements , the set of elements of g^∗ with compact stabilisers. This assumption on the image of the momentum map is equivalent to the assumption that M=GK_×N, for a compact Hamiltonian K-manifold N. The proof comes down to a reduction to the compact case. This reduction is based on a ‘quantisation commutes with induction’-principle, and involves a notion of induction of Hamiltonian group actions. This principle, in turn, is based on a version of the naturality of the assembly map for the inclusion K?G.     

Geometric quantization for proper actions

We first introduce an invariant index for G-equivariant elliptic differential operators on a locally compact manifold M admitting a proper cocompact action of a locally compact group G. It generalizes the Kawasaki index for orbifolds to the case of proper cocompact actions. Our invariant index is used to show that an analog of the Guillemin-Sternberg geometric quantization conjecture holds if M is symplectic with a Hamiltonian action of G that is proper and cocompact. This essentially solves a conjecture of Hochs and Landsman.     

On the structure of finite coverings of compact connected groups

Finite-sheeted covering mappings onto compact connected groups are studied. We show that for a covering mapping from a connected Hausdorff topological space onto a compact (in general, non-abelian) group there exists a topological group structure on the covering space such that the mapping becomes a homomorphism of groups. To prove this fact we construct an inverse system of covering mappings onto Lie groups which approximates the given covering mapping. As an application, it is shown that a covering mapping onto a compact connected abelian group G must be a homeomorphism provided that the character group of G admits division by degree of the mapping. We also get a criterion for triviality of coverings in terms of means and prove that each finite covering of G is equivalent to a polynomial covering.     

Minimal singular Riemannian foliations

We prove that a singular foliation on a compact manifold admitting an adapted Riemannian metric for which all leaves are minimal must be regular. To cite this article: V. Miquel, R.A. Wolak, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Harmonic forms and near-minimal singular foliations

For a closed 1-form with Morse singularities, Calabi discovered a simple global criterion for the existence of a Riemannian metric in which is harmonic. For a codimension 1 foliation , Sullivan gave a condition for the existence of a Riemannian metric in which all the leaves of are minimal hypersurfaces. The conditions of Calabi and Sullivan are strikingly similar. If a closed form has no singularities, then both criteria are satisfied and, for an appropriate choice of metric, is harmonic and the associated foliation is comprised of minimal leaves. However, when has singularities, the foliation is not necessarily minimal.? We show that the Calabi condition enables one to find a metric in which is harmonic and the leaves of the foliation are minimal outside a neighborhood U of the -singular set. In fact, we prove the best possible result of this type: we construct families of metrics in which, as U shrinks to the singular set, the taut geometry of the foliation outside U remains stable. Furthermore, all compact leaves missing U are volume minimizing cycles in their homology classes. Their volumes are controlled explicitly. Received: January 24, 2000     

A λ-lemma for foliations

We show that a C⁰ codimension one foliation with C¹ leaves F of a closed manifold is minimal if there are a foliation G transverse to F, and a diffeomorphism f preserving both foliations, such that every leaf of F intersects every leaf of G and f expands G. We use this result to study of Anosov actions on closed manifolds.     

The fixed energy problem for a class of nonconvex singular Hamiltonian systems

We consider a noncompact hypersurface H in R2N which is the energy level of a singular Hamiltonian of “strong force” type. Under global geometric assumptions on H, we prove that it carries a closed characteristic, as a consequence of a result by Hofer and Viterbo on the Weinstein conjecture in cotangent bundles of compact manifolds. Our theorem contains, as particular cases, earlier results on the fixed energy problem for singular Lagrangian systems of strong force type.     

À Propos D’un Théorème de J.-P. Jouanolou Concernant les Feuilles Fermées des Feuilletages Holomorphes

We generalize a theorem of Jouanolou and show that a codimension 1 holomorphic foliation (possibly singular) onany compact connected complex manifold has a finite number of closed leaves unless all leaves are closed.     

Induced Hamiltonian maps on the symplectic quotient

Let (M,ω) be a symplectic manifold and G a compact Lie group that acts on M. Assume that the action of G on M is Hamiltonian. Then a G-equivariant Hamiltonian map on M induces a map on the symplectic quotient of M by G. Consider an autonomous Hamiltonian H with compact support on M, with no non-constant closed trajectory in time less than 1 and time-1 map fH. If the map fH descends to the symplectic quotient to a map Φ(fH) and the symplectic manifold M is exact and Ham(M,ω) has no short loops, we prove that the Hofer norm of the induced map Φ(fH) is bounded above by the Hofer norm of fH.     

A class of vectorfields on S2 that are topologically equivalent to polynomial vectorfields

Let X be a C¹ vectorfield on S² = {(x, y, z)?R: x² + y² + z² = 1} such that no open subset of S² is the union of closed orbits of X. If X has only a finite number of singular orbits and satisfies one additional condition, then it is shown that X is topologically equivalent to a polynomial vectorfield. A corollary is that a foliation $\text{F}$ of the plane is topologically equivalent to a foliation by orbits of a polynomial vectorfield if and only if $\text{F}$ has only a finite number of inseparable leaves.     

G-Strands

A G-strand is a map g(t,s):?×?→G for a Lie group G that follows from Hamilton’s principle for a certain class of G-invariant Lagrangians. The SO(3)-strand is the G-strand version of the rigid body equation and it may be regarded physically as a continuous spin chain. Here, SO(3) _K -strand dynamics for ellipsoidal rotations is derived as an Euler–Poincaré system for a certain class of variations and recast as a Lie–Poisson system for coadjoint flow with the same Hamiltonian structure as for a perfect complex fluid. For a special Hamiltonian, the SO(3) _K -strand is mapped into a completely integrable generalization of the classical chiral model for the SO(3)-strand. Analogous results are obtained for the Sp(2)-strand. The Sp(2)-strand is the G-strand version of the Sp(2) Bloch–Iserles ordinary differential equation, whose solutions exhibit dynamical sorting. Numerical solutions show nonlinear interactions of coherent wave-like solutions in both cases. Diff(?)-strand equations on the diffeomorphism group G=Diff(?) are also introduced and shown to admit solutions with singular support (e.g., peakons).     

On smooth foliations with Morse singularities

Let M be a smooth manifold and let F be a codimension one, C^∞ foliation on M, with isolated singularities of Morse type. The study and classification of pairs (M,F) is a challenging (and difficult) problem. In this setting, a classical result due to Reeb (1946) [11] states that a manifold admitting a foliation with exactly two center-type singularities is a sphere. In particular this is true if the foliation is given by a function. Along these lines a result due to Eells and Kuiper (1962) [4] classifies manifolds having a real-valued function admitting exactly three non-degenerate singular points. In the present paper, we prove a generalization of the above mentioned results. To do this, we first describe the possible arrangements of pairs of singularities and the corresponding codimension one invariant sets, and then we give an elimination procedure for suitable center-saddle and some saddle-saddle configurations (of consecutive indices).In the second part, we investigate if other classical results, such as Haefliger and Novikov (Compact Leaf) theorems, proved for regular foliations, still hold true in presence of singularities. At this purpose, in the singular set, Sing(F) of the foliation F, we consider weakly stable components, that we define as those components admitting a neighborhood where all leaves are compact. If Sing(F) admits only weakly stable components, given by smoothly embedded curves diffeomorphic to S¹, we are able to extend Haefliger?s theorem. Finally, the existence of a closed curve, transverse to the foliation, leads us to state a Novikov-type result.     

Homoclinic orbits for a class of singular second order Hamiltonian systems in ?3

We consider a conservative second order Hamiltonian system $\ddot q + \nabla V(q) = 0$ in ?³ with a potential V having a global maximum at the origin and a line l ?? {0} = ? as a set of singular points. Under a certain compactness condition on V at infinity and a strong force condition at singular points we study, by the use of variational methods and geometrical arguments, the existence of homoclinic solutions of the system.