An inductive Lusternik-Schnirelmann category in a model category

We prove that the notion of an inductive category in a model category agrees with the Ganea approach given by Doeraene. This notion also coincides with the topological one when we consider the category of (well-)pointed topological spaces. An application is given.     

Lusternik-Schnirelmann category of a sphere-bundle over a sphere

We determine the Lusternik-Schnirelmann (L-S) category of a total space of a sphere-bundle over a sphere in terms of primary homotopy invariants of its characteristic map, and thus providing a complete answer to Ganea's Problem 4. As a result, we obtain a necessary and sufficient condition for a total space N to have the same L-S category as its ‘once punctured submanifold’ N\{P},P∈N. Also, necessary and sufficient conditions for a total space M to satisfy Ganea's conjecture are described.     

The proper L-S category of Whitehead manifolds

It is shown that the proper L-S category of an eventually end-irreducible, R²-irreducible Whitehead 3-manifold is 4. For this we prove, in the category of germs at infinity of proper maps, a partial analogue of the characterization by Eilenberg and Ganea of the L-S category of an aspherical space.     

On the Lusternik-Schnirelmann category of Stiefel manifolds

We determine the Lusternik-Schnirelmann category of real Stiefel manifolds Vn,k and quaternionic Stiefel manifolds Xn,k for n?2k which is equal to the cup-length of the mod 2 cohomology of Vn,k and the integer cohomology of Xn,k, respectively.     

The Lyusternik-Shnirel’man category, vector bundles, and immersions of manifolds

One of our two main results exhibits for a vector bundle over a compact Hausdorff spaceX an interplay between its span, its possible splittings, and the Lyusternik-Shnirel’man category ofX. The other main result, also on vector bundles and the Lyusternik-Shnirel’man category, enables us to derive certain inequalities connecting the immersion codimension, the stable span, and the Lyusternik-Shnirel’man category of a smooth closed manifold which is not stably parallelizable. Our results are applicable in various situations of general interest.     

Proper L-S category, fundamental pro-groups and 2-dimensional proper co-H-spaces

This paper presents a study of one-ended locally finite CW-complexes with proper L-S category ?2. We detect the class of towers of groups which can be the fundamental pro-group of a space of proper L-S category 2. A second part of the paper is concerned with two-dimensional CW-complexes. For these, we give different characterizations of spaces with proper L-S category ?2.     

Classifying foliations of 3-manifolds via branched surfaces

We use branched surfaces to define an equivalence relation on C¹ codimension one foliations of any closed orientable 3-manifold that are transverse to some fixed nonsingular flow. There is a discrete metric on the set of equivalence classes with the property that foliations that are sufficiently close (up to equivalence) share important topological properties.     

The Lyusternik–Shnirel'man category,¶vector bundles, and immersions of manifolds

One of our two main results exhibits for a vector bundle over a compact Hausdorff space X an interplay between its span, its possible splittings, and the Lyusternik–Shnirel'man category of X. The other main result, also on vector bundles and the Lyusternik–Shnirel'man category, enables us to derive certain inequalities connecting the immersion codimension, the stable span, and the Lyusternik–Shnirel'man category of a smooth closed manifold which is not stably parallelizable. Our results are applicable in various situations of general interest. Received: 4 September 1997     

Structure of foliations of codimension greater than one

We study the structure of a foliation of high codimension which admits a transverse foliation. We introduce four families of open saturated sets. These open sets have a simple characterization and allow us to establish a structure theorem as in codimension 1.     

Affine foliations and covering hyperbolic structures

We describe the relationship between closed affine laminations in a punctured surface and some associated hyperbolic structures on certain covers of the punctured surface, which we call covering hyperbolic structures. Further, in analogy with the theory of William Thurston relating the Teichmüller space of a surface to the projective lamination space, we describe a space with points representing affine laminations in a given surface and with other points representing the associated covering hyperbolic structures. Received: 27 March 2000 / Revised version: 10 January 2001     

Smooth actions of finite Oliver groups on spheres

In this article, we deal with the following two questions. For smooth actions of a given finite group G on spheres S, which smooth manifolds F occur as the fixed point sets in S, and which real G-vector bundles ν over F occur as the equivariant normal bundles of F in S? We focus on the case G is an Oliver group and answer both questions under some conditions imposed on G, F, and ν. We construct smooth actions of G on spheres by making use of equivariant surgery, equivariant thickening, and Oliver's equivariant bundle extension method modified by an equivariant wegde sum construction and an equivariant bundle subtraction procedure.     

L-S category of quaternionic Stiefel manifolds

By calculating certain generalized cohomology theory, lower bounds for the L-S category of quaternionic Stiefel manifolds are given.     

Invariants of transverse foliations

We construct two invariants for a pair of transverse one-dimensional foliations on the plane. If the set of separatrices is Hausdorff in the space of leaves, the invariant is a distinguished graph. In case there are a finite number of separatrices the invariant is an indexed link.     

On elliptic modular foliations

In this article we consider the three parameter family of elliptic curves E^t: y² − 4(x − t₁ )³ + t² (x − t₁) + t₃ = 0, t ∈ ?³, and study the modular holomorphic foliation ?ω, in ?³ whose leaves are constant locus of the integration of a l-form ω over topological cycles of E_t. Using the Gauss—Manin connection of the family E^t, we show that ?ω is an algebraic foliation. In the case , we prove that a transcendent leaf of ?ω contains at most one point with algebraic coordinates and the leaves of ?ω corresponding to the zeros of integrals, never cross such a point. Using the generalized period map associated to the family E^t, we find a uniformization of ?ω in T, where T ⊂ ?³ is the locus of parameters t for which E_t is smooth. We find also a real first integral of ?ω. restricted to T and show that ?ω is given by the Ramanujan relations between the Eisenstein series.     

A Quillen model category structure on some categories of comonoids

We prove that for certain monoidal (Quillen) model categories, the category of comonoids therein also admits a model structure.