Ganea and Whitehead definitions for the tangential Lusternik-Schnirelmann category of foliations

This work solves the problem of elaborating Ganea and Whitehead definitions for the tangential category of a foliated manifold. We develop these two notions in the category \Tops of stratified spaces, that are topological spaces $X$ endowed with a partition \cF and compare them to a third invariant defined by using open sets. More precisely, these definitions apply to an element (X,\cF) of \Tops together with a class \cA of subsets of $X$; they are similar to invariants introduced by M. Clapp and D. Puppe. If (X,\cF)\in\Tops, we define a transverse subset as a subspace $A$ of $X$ such that the intersection $S\cap A$ is at most countable for any S\in \cF. Then we define the Whitehead and Ganea LS-categories of the stratified space by taking the infimum along the transverse subsets. When we have a closed manifold, endowed with a $C^1$-foliation, the three previous definitions, with \cA the class of transverse subsets, coincide with the tangential category and are homotopical invariants.Comment: 14 pages, 2 figure

Transverse Lusternik–Schnirelmann category of foliated manifolds

AbstractThe purpose of this paper is to develop a transverse notion of Lusternik–Schnirelmann category in the field of foliations. Our transverse category, denoted by cat⋔(M,F), is an invariant of the foliated homotopy type which is finite on compact manifolds. It coincides with the classical notion when the foliation is by points. We prove that for any foliated manifold catM⩽catLcat⋔(M,F), where L is a leaf of maximal category, thus generalizing a result of Varadarajan for fibrations. Also we prove that cat⋔(M,F) is bounded below by the index of k∗Hb+(M), the latter being the image in HDR(M) of the algebra of basic cohomology in positive degrees. In the second part of the paper we prove that cat⋔(M,F) is a lower bound for the number of critical leaves of any basic function provided that F is a foliation satisfying certain conditions of Palais–Smale type. As a consequence, we prove that the result is true for compact Hausdorff foliations and for foliations of codimension one. This generalizes the classical result of Lusternik and Schnirelmann about the number of critical points of a smooth function

Minimal foliations on lie groups

AbstractLet F(G,H) be the foliation determined by a (not necessarily closed) Lie subgroup H of a Lie group G. We prove that there always exists a Riemannian metric on G for which the leaves of F(G,H) are minimal submanifolds