Foliations invariant under Lie group transverse actions

In this paper we study (smooth and holomorphic) foliations which are invariant under transverse actions of Lie groups. Authors’ address: Alexandre Behague and Bruno Scárdua, Instituto de Matemática, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, 21945-970 Rio de Janeiro, RJ, Brazil     

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.     

Analogies between group actions on 3-manifolds and number fields

Let a cyclic group $G$ act either on a number field $\mathbb L$ or on a $3$-manifold $M$. Let $s_{\mathbb L}$ be the number of ramified primes in the extension $\mathbb L^G\subset \mathbb L$ and $s_M$ be the number of components of the branching set of the branched covering $M\to M/G$. In this paper, we prove several formulas relating $s_{\mathbb L}$ and $s_M$ to the induced $G$-action on $Cl(\mathbb L)$ and $H_1(M),$ respectively. We observe that the formulas for $3$-manifolds and number fields are almost identical, and therefore, they provide new evidence for the correspondence between $3$-manifolds and number fields postulated in arithmetic topology.     

Characterizations of almost periodic homeomorphisms

In this paper we study recurrent and almost periodic homeomorphisms on the Euclidean space Rm; we give conditions under which recurrent implies periodic. On the other hand we give properties of elements of compact groups of diffeomorphisms on manifolds.     

On singular foliations on the solid torus

We study smooth foliations on the solid torus S¹×D²$S^1\times D^2$ having S¹×{0}$S^1\times \{\mathbf{0}\}$ and S¹×∂D²$S^1\times \partial D^2$ as the only compact leaves and S¹×{0}$S^1\times \{\mathbf{0}\}$ as singular set. We show that all other leaves can only be cylinders or planes, and give necessary conditions for the foliation to be a suspension of a diffeomorphism of the disc.     

Diffeomorphism classification of finite group actions on disks

In this paper we discuss the diffeomorphism classification of finite group actions on disks. We answer the question when an action on a space M can be extended to an action on a disk such that the action is free away from M. Let the singular set consist of the points with nontrivial isotropy group. We show (under some dimension assumptions) that disks with diffeomorphic neighborhoods of the singular set can be imbedded into each other. As a consequence we find a classification of group actions on disks in terms of the neighborhood of the singular set and an element in the Whitehead group of G.     

Nonsmoothable group actions on elliptic surfaces

Let G be a cyclic group of order 3, 5 or 7, and X=E(n) be the relatively minimal elliptic surface with rational base. In this paper, we prove that under certain conditions on n, there exists a locally linear G-action on X which is nonsmoothable with respect to infinitely many smooth structures on X. This extends the main result of [X. Liu, N. Nakamura, Pseudofree Z/3-actions on K3 surfaces, Proc. Amer. Math. Soc. 135 (3) (2007) 903-910].     

Three-dimensional Lie group actions on compact (4n+3)-dimensional geometric manifolds

The (4n+3)-dimensional sphere S⁴ⁿ⁺³ can be viewed as the boundary of the quaternionic hyperbolic space and the group PSp(n+1,1) of quaternionic hyperbolic isometries extends to a real analytic transitive action on S⁴ⁿ⁺³. We call the pair (PSp(n+1,1),S⁴ⁿ⁺³) a spherical Q C-C geometry. A manifold M locally modelled on this geometry is said to be a spherical Q C-C manifold. We shall classify all pairs (G,M) where G is a three-dimensional connected Lie group which acts smoothly and almost freely on a compact spherical Q C-C manifold M, preserving the geometric structure. As an application, we shall determine all compact 3-pseudo-Sasakian manifolds admitting spherical Q C-C structures.     

On Reeb components of invariant foliations of projectively Anosov flows

We show that if a C² codimension one foliation on a three-dimensional manifold has a Reeb component and is invariant under a projectively Anosov flow, then it must be a Reeb foliation on S²×S¹.     

Signatures of foliated surface bundles and the symplectomorphism groups of surfaces

For any closed oriented surface Σ_g of genus g?3, we prove the existence of foliatedΣ_g-bundles over surfaces such that the signatures of the total spaces are non-zero. We can arrange that the total holonomy of the horizontal foliations preserve a prescribed symplectic form ω on the fiber. We relate the cohomology class represented by the transverse symplectic form to a crossed homomorphism which is an extension of the flux homomorphism from the identity component to the whole group of symplectomorphisms of Σ_g with respect to the symplectic form ω.     

Local structure of Koszul-Vinberg and of Lie algebroids

In this short note we continue our study of Koszul-Vinberg algebroids which form a subcategory of the category of Lie algebroids, and which appear naturally in the study of affine structures, affine and transversally affine foliations [N. Nguiffo Boyom, R. Wolak, J. Geom. Phys. 42 (2002) 307-317]. We prove a local decomposition theorem for KV-algebroids. Using the notion of KV-algebroids we introduce a new class of singular foliations: affine singular foliations. In the last section we study the holonomy of these foliations and prove a stability theorem.     

Multivariate lag-windows and group representations

Symmetries of the auto-cumulant function (a generalization of the auto-covariance function) of a kth-order stationary time series are derived through a connection with the symmetric group of degree k. Using the theory of group representations, symmetries of the auto-cumulant function are demystified and lag-window functions are symmetrized to satisfy these symmetries. A generalized Gabr–Rao optimal kernel is also derived through the developed theory.     

Topology and the robot arm

A robot arm is in effect a smooth function from the space of positions of the arm to the space of positions of a coordinate frame attached to the end of the arm. For the most common robots built today, this means a map f: T ⁿ R ³×SO ₃. We describe the singularities of this map. The set of rotational singularities is the set of arm positions where the axes of the links are parallel to a plane. Thus, it is always two-dimensional. Also, we show that f is homotopic to a map which factors through a circle, and represents the generator of ₁(SO ₃). The engineering implication of these statements are discussed.     

Free circle actions with contractible orbits on symplectic manifolds

We prove that closed symplectic four-manifolds do not admit any smooth free circle actions with contractible orbits, without assuming that the actions preserve the symplectic forms. In higher dimensions such actions by symplectomorphisms do exist, and we give explicit examples based on the constructions of FGM.     

Dimensions of fixed point sets of involutions

Suppose the fixed point set F of a smooth involution T:M → M on a smooth, closed and connected manifold M decomposes into two components Fⁿ and F² of dimensions n and 2, respectively, with n > 2 odd. We show that the codimension k of Fⁿ is small if the normal bundle of F² does not bound; specifically, we show that k≦ 3 in this case. In the more general situation where F is not a boundary, n (not necessarily odd) is the dimension of a component of F of maximal dimension and k is the codimension of this component, and fixed components of all dimensions j, 0≦ j≦ n, may occur, a theorem of Boardman gives that . In addition, we show that this bound can be improved to k≦ 1 (hence k = 1) for some specific values of n and some fixed stable cobordism classes of the normal bundle of F² in M; further, we determine in these cases the equivariant cobordism class of (M, T). Received: 25 August 2005     

Finite group actions on bordered surfaces of small genus

This paper considers finite group actions on compact bordered surfaces — quotients of unbordered orientable surfaces under the action of a reflectional symmetry. Classification of such actions (up to topological equivalence) is carried out by means of the theory of non-euclidean crystallographic groups, and determination of normal subgroups of finite index in these groups, up to conjugation within their automorphism group. A result of this investigation is the determination, up to topological equivalence, of all actions of groups of finite order 6 or more on compact (orientable or non-orientable) bordered surfaces of algebraic genus p for 2≤p≤6. We also study actions of groups of order less than 6, or of prime order, on bordered surfaces of arbitrary algebraic genus p≥2.